QDPLFV RI 1HZ 9HQWXUHV - Haas School of · PDF file9doxdwlrq dqg 5hwxuq ’\qdplfv ri 1hz...

$: QDPLFV RI 1HZ 9HQWXUHV - Haas School of · PDF file9doxdwlrq dqg 5hwxuq ’\qdplfv ri 1hz 9hqwxuhv -rqdwkdq % %hun +ddv 6fkrro ri %xvlqhvv ... /hw9>q w ˚p w ˚p w @ehwkhfxuuhqwydoxhriwkhnupfrqglwlrqdorq$
9DOXDWLRQ DQG 5HWXUQ '\QDPLFV RI 1HZ 9HQWXUHV�

-RQDWKDQ %� %HUN+DDV 6FKRRO RI %XVLQHVV

8QLYHUVLW\ RI &DOLIRUQLD� %HUNHOH\DQG

1%(5(PDLO� EHUN#X�ZDVKLQJWRQ�HGX

5LFKDUG &� *UHHQ*UDGXDWH 6FKRRO RI ,QGXVWULDO $GPLQLVWUDWLRQ

&DUQHJLH 0HOORQ 8QLYHUVLW\(PDLO� UJ�E#DQGUHZ�FPX�HGX

DQG

9DVDQW 1DLN)DFXOW\ RI &RPPHUFH

8QLYHUVLW\ RI %ULWLVK &ROXPELDDQG

/HKPDQ %URWKHUV(PDLO� YDVDQW#nQDQFH�FRPPHUFH�XEF�FD

6HSWHPEHU ��

�:H ZLVK WR WKDQN /RUHQ]R *DUODSSL� :DOWHU 1RYDHV� %U\DQ 5RXWOHGJH� -Hm 7DUORZH� DQG 7DQ :DQJIRU YDOXDEOH FRPPHQWV DQG VXJJHVWLRQV� :H DFNQRZOHGJH JUDWHIXOO\ UHVHDUFK VXSSRUW IURP WKH 4�*URXS�WKH 'HDQV 5HVHDUFK )XQG DQG WKH &HQWHU IRU )LQDQFLDO 0DUNHWV DW &DUQHJLH 0HOORQ �*UHHQ�� DQG WKH(QWUHSUHQHXUVKLS 5HVHDUFK $OOLDQFH DW 8%& DQG 66+5& �1DLN��

$EVWUDFW

9DOXDWLRQ DQG 5HWXUQ '\QDPLFV RI 5' 9HQWXUHV

:H GHYHORS DQG DQDO\]H D PRGHO RI D PXOWL�VWDJH LQYHVWPHQW SURMHFW WKDW FDSWXUHV PDQ\ IHDWXUHVRI 5' YHQWXUHV DQG VWDUW�XS FRPSDQLHV� $Q LPSRUWDQW IHDWXUH WKHVH SUREOHPV VKDUH LV WKDW WKHnUP OHDUQV DERXW WKH SRWHQWLDO SURnWDELOLW\ RI WKH SURMHFW WKURXJKRXW LWV OLIH� EXW WKDW ?WHFKQLFDOXQFHUWDLQW\� DERXW WKH UHVHDUFK DQG GHYHORSPHQW HmRUW LWVHOI LV RQO\ UHVROYHG WKURXJK DGGLWLRQDOLQYHVWPHQW E\ WKH nUP� ,Q DGGLWLRQ� WKH ULVNV DVVRFLDWHG ZLWK WKH XOWLPDWH FDVK oRZV WKH nUPUHDOL]HV RQ FRPSOHWLRQ RI WKH SURMHFW KDYH D V\VWHPDWLF FRPSRQHQW� ZKLOH WKH SXUHO\ WHFKQLFDO ULVNVDUH LGLRV\QFUDWLF� 2XU PRGHO FDSWXUHV WKHVH GLmHUHQW VRXUFHV RI ULVN� DQG DOORZV XV WR VWXG\ WKHLULQWHUDFWLRQ LQ GHWHUPLQLQJ WKH ULVN SUHPLD HDUQHG E\ WKH YHQWXUH GXULQJ GHYHORSPHQW� 2XU UHVXOWVVKRZ WKDW WKH V\VWHPDWLF ULVN� DQG WKH UHTXLUHG ULVN SUHPLXP� RI WKH YHQWXUH DUH KLJKHVW HDUO\ LQLWV OLIH� DQG GHFUHDVH DV LW DSSURDFKHV FRPSOHWLRQ� GHVSLWH WKH LGLRV\QFUDWLF QDWXUH RI WKH WHFKQLFDOULVN�

� ,QWURGXFWLRQ

7KH YDOXDWLRQ RI YHQWXUHV WKDW SURGXFH FDVK oRZV RQO\ DIWHU PXOWLSOH VWDJHV RI GHYHORSPHQW LV

DQ LPSRUWDQW SUREOHP IRU nQDQFLDO DQDO\VLV� :LWKLQ HVWDEOLVKHG RUJDQL]DWLRQV� PDQDJHUV DUH XQGHU

LQFUHDVLQJ SUHVVXUH WR TXDQWLWDWLYHO\ MXVWLI\ H[SHQGLWXUHV RQ 5'� )LQDQFLDO LQWHUPHGLDULHV� VXFK DV

YHQWXUH FDSLWDOLVWV� URXWLQHO\ DOORFDWH FDSLWDO UHVRXUFHV RQ WKH EDVLV RI LPSOLFLW RU H[SOLFLW YDOXDWLRQV

RI nUPV ZLWK QR DVVHWV� RU IHZ DVVHWV� RWKHU WKDQ WKH SRWHQWLDO WR JHQHUDWH FDVK oRZV LQ WKH IXWXUH�

6XFK nUPV DUH D JURZLQJ SUHVHQFH RQ RUJDQL]HG HTXLW\ PDUNHWV DV ZHOO�

5' YHQWXUHV DUH VXEMHFW WR VHYHUDO� TXDOLWDWLYHO\ GLmHUHQW VRXUFHV RI ULVN� 7KHUH LV WKH ?WHFK�

QLFDO� XQFHUWDLQW\ DVVRFLDWHG ZLWK WKH VXFFHVV RI WKH UHVHDUFK LWVHOI� RU WKH WLPH DQG FRVW UHTXLUHG

WR EULQJ LW WR IUXLWLRQ� ([DPSOHV DUH WKH FOLQLFDO WULDOV RI D QHZ GUXJ� RU H[SORUDWRU\ GULOOLQJ IRU

RLO� 7KHUH LV WKH H[RJHQRXV ULVN DVVRFLDWHG ZLWK WKH DFWLRQV RI D FRPSHWLWRU RU FKDQJHV LQ WKH

HQYLURQPHQW EHIRUH RU VRRQ DIWHU SURGXFW LQWURGXFWLRQ� )RU H[DPSOH� D QHZ GUXJ PD\ EH UHQGHUHG

XQQHFHVVDU\ E\ D VXSHULRU WUHDWPHQW RSWLRQ� $ VRIWZDUH SURGXFW PD\ IDLO EHFDXVH RI WHFKQRORJLFDO

DGYDQFHV LQ KDUGZDUH� )LQDOO\� WKHUH DUH WKH PRUH WUDGLWLRQDO ULVNV DVVRFLDWHG ZLWK GHPDQG IRU WKH

SURGXFW DQG SURGXFWLRQ FRVWV�

+RZ GR WKHVH W\SHV RI ULVN MRLQWO\ GHWHUPLQH WKH YDOXH RI DQ 5' YHQWXUH" +RZ PXFK V\VWHPDWLF

YHUVXV XQV\VWHPDWLF ULVN LV ERUQH E\ LQYHVWRUV LQ 5'� DQG KRZ GRHV WKLV LQoXHQFH WKH ULVN SUHPLXP

WKH\ UHTXLUH" ,Q WKLV SDSHU ZH GHYHORS D PRGHO WKDW DOORZV IRU DOO WKUHH W\SHV RI XQFHUWDLQW\� DQG LV

VWLOO VXpFLHQWO\ WUDFWDEOH WR SURYLGH TXDOLWDWLYH DQG TXDQWLWDWLYH JXLGDQFH UHJDUGLQJ WKHVH TXHVWLRQV�

7H[WERRNV LQ FRUSRUDWLRQ nQDQFH� LQ UHVSRQVH WR WKH WHQGHQF\ RI SUDFWLWLRQHUV DQG VWXGHQWV WR

FRQIXVH V\VWHPDWLF DQG XQV\VWHPDWLF ULVNV� RIWHQ SRLQW WR WKH WHFKQLFDO ULVNV RI 5' DV H[DPSOHV

RI GLYHUVLnDEOH ULVNV WKDW GR QRW UHTXLUH FRPSHQVDWLRQ� %UHDOH\ DQG 0\HUV �� IRU LQVWDQFH�

OLVW WKH ULVN RI D GU\ KROH LQ RLO H[SORUDWLRQ� DQG GHQLDO RI DSSURYDO IRU D GUXJ WKDW FXUHV EDOGQHVV

E\ WKH )RRG DQG 'UXJ $GPLQLVWUDWLRQ DV H[DPSOHV RI ?EDG RXWFRPHV�� WKDW ?DSSHDU WR UHoHFW

XQLTXH �L�H�� GLYHUVLnDEOH� ULVNV ZKLFK ZRXOG QRW DmHFW WKH H[SHFWHG UDWH RI UHWXUQ GHPDQGHG E\

LQYHVWRUV� �S� �� WKHLU SDUHQWKHVHV�� %UHDOH\� 0\HUV DQG 0DUFXV �� DJDLQ FRQVLGHULQJ WKH

H[DPSOH RI RLO H[SORUDWLRQ� DUJXH ?� � � IRU WKH FRPSDQ\ DV D ZKROH� LWV WKH DYHUDJH VXFFHVV UDWH WKDW

PDWWHUV� *HRORJLFDO ULVNV �LV WKHUH RLO RU QRW"� VKRXOG DYHUDJH RXW� 7KH ULVN RI D ZRUOGZLGH GULOOLQJ

SURJUDP LV PXFK OHVV WKDQ WKH DSSDUHQW ULVN RI DQ\ VLQJOH ZLOGFDW ZHOO� �S� �� WKHLU HPSKDVLV DQG

SDUHQWKHVHV�� 7KH\ JR RQ WR VD\ WKDW WKH LQYHVWRUV LQ DQ RLO FRPSDQ\V VWRFN ZLOO ?� � � QDWXUDOO\_

DQG UHDOLVWLFDOO\_DVVXPH WKDW \RXU VXFFHVVHV DQG IDLOXUHV LQ GULOOLQJ RLO ZHOOV ZLOO DYHUDJH RXW ZLWK

WKH WKRXVDQGV RI LQGHSHQGHQW EHWV PDGH E\ WKH FRPSDQLHV LQ WKHLU SRUWIROLR� �S� ��

7KHVH DUJXPHQWV PLJKW EH FRQVWUXHG WR LPSO\ WKDW 5' SURMHFWV ZRXOG QRW UHTXLUH D SDUWLFX�

ODUO\ KLJK H[SHFWHG UHWXUQ� ,Q RXU PRGHO ZH YDOXH VLPXOWDQHRXVO\ D PXOWL�VWDJH 5' SURMHFW DQG

WKH VWUHDP RI FDVK oRZV LW ZLOO GHOLYHU LI VXFFHVVIXO� :H nQG WKH UHTXLUHG ULVN SUHPLXP IRU WKH 5'

LV KLJKHU WKDQ WKDW RI WKH XQGHUO\LQJ FDVK oRZV� DQG WKDW WKH ULVN SUHPLXP GHFUHDVHV DV WKH SURMHFW

DSSURDFKHV FRPSOHWLRQ� 7KLV EHKDYLRU HPHUJHV GHVSLWH WKH LGLRV\QFUDWLF QDWXUH RI WKH WHFKQLFDO ULVN

DVVRFLDWHG ZLWK FRPSOHWLQJ WKH 5'� :H DUJXH WKDW UHDVRQLQJ EDVHG VLPSO\ RQ WKH LGLRV\QFUDWLF

�

QDWXUH RI SXUHO\ WHFKQLFDO ULVN LJQRUHV DQ LPSRUWDQW IHDWXUH RI DFWXDO 5' SURMHFWV� 'HFLVLRQV WR

FRQWLQXH ZLWK 5' DUH PDGH FRQGLWLRQLQJ RQ WKH UHVROXWLRQ RI V\VWHPDWLF DV ZHOO DV XQV\VWHP�

DWLF XQFHUWDLQW\� )RU H[DPSOH� ZKLOH ?JHRORJLFDO ULVNV� DUH DUJXDEO\ GLYHUVLnDEOH� WKH GHFLVLRQ WR

FRQWLQXH GULOOLQJ DW DQ\ SDUWLFXODU VLWH� DV HDFK VWDJH RI WKH GHYHORSPHQW SURFHVV LV FRPSOHWHG DQG

WHFKQLFDO XQFHUWDLQW\ LV UHVROYHG� LV PDGH ZLWK WKH FXUUHQW SULFH RI RLO LQ PLQG� 7KLV LPSDUWV WR WKH

SURMHFW WKH FKDUDFWHULVWLFV RI D FRPSRXQG RSWLRQ� 2SWLRQV� EHFDXVH RI WKH LPSOLFLW OHYHUDJH WKH\

LPSDUW� KDYH KLJKHU V\VWHPDWLF ULVN WKDQ WKH XQGHUO\LQJ DVVHW�

,Q RXU PRGHO WKH nUP FRQVLVWV RI D VLQJOH 5' SURMHFW� :KHQ WKH 5' LV VXFFHVVIXOO\ FRPSOHWHG�

WKH nUP ZLOO JHQHUDWH D VWUHDP RI VWRFKDVWLF FDVK oRZV� ZKLFK ZH PRGHO DV D JHRPHWULF UDQGRP

ZDON WKDW KDV ERWK LGLRV\QFUDWLF DQG V\VWHPDWLF ULVN� ,Q RUGHU WR UHDOL]H WKHVH FDVK oRZV� KRZHYHU�

WKH nUP PXVW nUVW FRPSOHWH 1 VWDJHV RI 5'� ,Q WKLV SURFHVV� WKH YHQWXUH LV VXEMHFW WR IRXU W\SHV

RI ULVN�

�� 7HFKQLFDO ULVNV SHUWDLQLQJ WR WKH VXFFHVVIXO FRPSOHWLRQ RI WKH 5' LWVHOI� $W HDFK GDWH DQ

LQYHVWPHQW LV UHTXLUHG LQ RUGHU WR NHHS WKH SURMHFW ?DFWLYH�� ,I WKH LQYHVWPHQW LV PDGH� WKHQ

ZLWK VRPH SUREDELOLW\ WKH FXUUHQW VWDJH ZLOO EH VXFFHVVIXOO\ FRPSOHWHG DQG WKH nUP FDQ PRYH

RQ WR WKH QH[W VWDJH� 7KH HYHQW RI VXFFHVV RU IDLOXUH DW HDFK VWDJH LV LGLRV\QFUDWLF ULVN�

�� 7KH ULVN RI FRPSHWLWLYH WKUHDW RU REVROHVFHQFH� :H DVVXPH WKDW ZLWK D n[HG SUREDELOLW\ LQ

HDFK SHULRG WKH FDVK oRZV IURP WKH SURMHFW� ZKHWKHU DFWXDO RU SRWHQWLDO� DUH H[WLQJXLVKHG�

7KH XQFHUWDLQW\ UHJDUGLQJ REVROHVFHQFH LV LGLRV\QFUDWLF�

�� 8QFHUWDLQW\ DVVRFLDWHG WKH SRWHQWLDO IXWXUH FDVK oRZV WKH SURMHFW ZLOO SURGXFH LI FRPSOHWHG�

$W DQ\ SRLQW LQ WLPH WKH nUP REVHUYHV WKH FDVK oRZV WKH SURMHFW ZRXOG EH SURGXFLQJ LI LW

ZHUH DOUHDG\ FRPSOHWH� DQG WKLV VWDWH YDULDEOH KDV ERWK V\VWHPDWLF DQG LGLRV\QFUDWLF ULVN�

�� /HDUQLQJ E\ GRLQJ� 7KH SUREDELOLW\ RI VXFFHVV DW HDFK VWDJH LV XQNQRZQ WR WKH nUPV PDQ�

DJHPHQW� ZKR� EDVHG RQ SDVW RXWFRPHV� XSGDWH WKHLU EHOLHIV DERXW LW� 8QFHUWDLQW\ DERXW

WKH GXUDWLRQ DQG FRVW RI WKH 5' LV WKXV UHVROYHG E\ PDNLQJ LQYHVWPHQWV LQ WKH SURFHVV

DQG REVHUYLQJ WKH RXWFRPHV� 7KH UHVXOWLQJ H[SHFWHG FRVW WR FRPSOHWLRQ IRU WKH 5' HYROYHV

HQGRJHQRXVO\� UHoHFWLQJ RSWLPDO SROLFLHV�

,Q PDNLQJ LWV GHFLVLRQV DERXW ZKHWKHU WR FRQWLQXH LQYHVWLQJ� ?PRWKEDOO� WKH SURMHFW� RU VLPSO\

DEDQGRQ WKH YHQWXUH� WKH nUP REVHUYHV VWDWH YDULDEOHV VXPPDUL]LQJ HDFK RI WKH XQGHUO\LQJ VRXUFHV

RI XQFHUWDLQW\ DERYH� �� WKH QXPEHU RI VWDJHV RI 5' VWLOO WR EH FRPSOHWHG� �� ZKHWKHU WKH

SURMHFWV SRWHQWLDO FDVK oRZV KDYH EHHQ H[WLQJXLVKHG WKURXJK REVROHVFHQFH� �� WKH OHYHO RI FDVK

oRZ WKH SURMHFW ZRXOG EH HDUQLQJ LI WKH 5' ZHUH FRPSOHWH� DQG �� WKH KLVWRU\ RI LWV SDVW DWWHPSWV

DW FRPSOHWLRQ�

:H DUH DEOH WR REWDLQ D QXPEHU RI LQWHUHVWLQJ UHVXOWV RQ WKH LQWHUDFWLRQ RI WKHVH GLmHUHQW VRXUFHV

RI ULVN LQ GHWHUPLQLQJ WKH YDOXH DQG H[SHFWHG UHWXUQ RQ WKH YHQWXUH� 3HUKDSV PRVW VXUSULVLQJ LV WKH

UROH WKH UHVROXWLRQ RI LGLRV\QFUDWLF XQFHUWDLQW\ SOD\V LQ GHWHUPLQLQJ WKH FXUUHQW ULVN SUHPLXP� $V

WHFKQLFDO �LGLRV\QFUDWLF� XQFHUWDLQW\ LV UHVROYHG� WKH SUREDELOLW\ RI VXFFHVVIXO DQG WLPHO\ FRPSOHWLRQ

�

FKDQJHV� 7KLV FKDQJHV WKH SURSHUWLHV RI WKH RSWLRQ WR PRWKEDOO WKH SURMHFW� ZKLFK LV H[HUFLVHG ZLWK

WKH SRWHQWLDO IXWXUH FDVK oRZV LQ PLQG� 7KXV� DOWKRXJK WKHUH LV QR ULVN SUHPLXP HDUQHG RQ

LGLRV\QFUDWLF ULVN SHU VH� WKH UHVROXWLRQ RI LGLRV\QFUDWLF XQFHUWDLQW\ FDQ GUDPDWLFDOO\ DOWHU WKH ULVN

SUHPLXP HDUQHG RQ WKH YHQWXUH DV D ZKROH�

:H DUH DOVR DEOH WR GHVFULEH XSSHU DQG ORZHU ERXQGV RQ WKH ULVN SUHPLXP RI WKH 5'� DQG

WKXV RQ WKH FRVW RI FDSLWDO RI ?SXUH JURZWK� nUPV� :H GHULYH DQ DQDO\WLFDOO\ VWUDLJKWIRUZDUG XSSHU

ERXQG RQ WKH ULVN SUHPLXP� 7KH ULVN SUHPLXP RI WKH 5' LV FORVH WR WKLV XSSHU ERXQG ZKHQ WKH

SURMHFW LV PRWKEDOOHG� JHQHUDOO\ HDUO\ LQ LWV OLIH� :H VKRZ QXPHULFDOO\ WKDW WKH ULVN SUHPLXP RI WKH

5' ZLOO QRW EH ORZHU WKDQ WKH SUHPLXP HDUQHG RQ WKH FDVK oRZV IURP WKH SURMHFW RQ FRPSOHWLRQ�

7KH ULVN SUHPLXP RQ WKH 5' SURMHFW IDOOV IURP WKH XSSHU WR WKH ORZHU ERXQG RYHU LWV OLIH� 7KLV

GHFOLQH LQ WKH ULVN SUHPLXP LV QRQ�OLQHDU� DQG LV SDUWLFXODUO\ SURQRXQFHG DURXQG WKH WLPH WKH nUP

MXVW EHJLQV DFWLYH GHYHORSPHQW�

/HDUQLQJ E\ GRLQJ DOVR SOD\V DQ LPSRUWDQW UROH LQ GHWHUPLQLQJ WKH SDWWHUQV RI UHWXUQV HDUQHG

RYHU WKH OLIH RI WKH SURMHFW� $V D UHVXOW RI OHDUQLQJ� DQG HQGRJHQRXV SROLF\ UHVSRQVHV WR LW� HDUO\

UHVROXWLRQ RI WHFKQLFDO XQFHUWDLQW\ KDV D PXFK JUHDWHU LPSDFW RQ YDOXH DQG H[SHFWHG UHWXUQ WKDQ

WHFKQLFDO VXFFHVV ODWHU LQ WKH SURMHFWV OLIH� 7KH nUP ZLOO WROHUDWH RQO\ D FHUWDLQ QXPEHU RI IDLOXUHV

DW D JLYHQ VWDJH RI GHYHORSPHQW EHIRUH LW DEDQGRQV WKH 5'� 7KLV WKUHVKROG LV TXLWH ORZ LQ WKH

HDUO\ VWDJHV� VR LW LV TXLWH SRVVLEOH IRU WZR LGHQWLFDO nUPV WR ZRUN RQ LGHQWLFDO SURMHFWV ZLWK YHU\

GLmHUHQW UHVXOWV� 2QH nUP PLJKW EH OXFN\� LQ WKDW LW DFKLHYHV SRVLWLYH UHDOL]DWLRQV RI XQFHUWDLQW\ LQ

WKH HDUO\ VWDJHV DQG XOWLPDWHO\ FRPSOHWHV WKH SURMHFW� 7KH RWKHU nUP� EHFDXVH RI EDG OXFN HDUO\

RQ� RSWLPDOO\ GHFLGHV WR DEDQGRQ WKH SURMHFW DQG QHYHU OHDUQV DERXW LWV XQGHUO\LQJ SURnWDELOLW\�

7KURXJK VLPXODWLRQ ZH DUH DEOH WR GHVFULEH WKH SURSHUWLHV RI UHDOL]HG UHWXUQV� :H VKRZ WKDW WKH

RSWLRQ WR DEDQGRQ FRPELQHV ZLWK WKH WHFKQRORJLFDO XQFHUWDLQW\ WR SURGXFH ELPRGDO GLVWULEXWLRQV�

DV RQH PLJKW H[SHFW RI KLJK ULVN� WHFKQRORJLFDO YHQWXUHV� 7KH H[SHFWHG FDVK oRZV IURP D W\SLFDO

SURMHFW VKRZ WKH OLIH F\FOH HmHFWV WKDW FKDUDFWHUL]H QHZ SURGXFWV RU LQGXVWULHV�

:KLOH VLPSOH� RXU PRGHO RI 5' LV DOVR oH[LEOH� 6WDJHV RI GLmHUHQW OHQJWK� RU ZLWK DV\PPHWULF

SUREDELOLWLHV RI VXFFHVV� FDQ WR VRPH H[WHQW EH PRGHOHG DV VHTXHQFHV RI VWDJHV� :H REWDLQ FORVHG�

IRUP VROXWLRQV IRU VRPH VSHFLDO FDVHV� ZKLFK SURYLGH D EHQFKPDUN DJDLQVW ZKLFK WR HYDOXDWH WKH

QXPHULFDO VROXWLRQV ZH VWXG\ IRU WKH JHQHUDO PRGHO� :H DUH DOVR DEOH WR SURYH D QXPEHU RI UHVXOWV

FKDUDFWHUL]LQJ VROXWLRQV WR WKH PRUH JHQHUDO FDVHV� HYHQ ZKHUH ZH DUH XQDEOH WR VROYH LQ FORVHG�IRUP

IRU RSWLPDO SROLFLHV DQG YDOXHV�

,Q WKH QH[W VHFWLRQ ZH GLVFXVV RWKHU SDSHUV LQ WKH OLWHUDWXUH UHODWHG WR RXUV� 6HFWLRQ � LQWURGXFHV

WKH QRWDWLRQ DQG GHVFULEHV WKH VWUXFWXUH RI WKH PRGHO� 6HFWLRQ � WKHQ SUHVHQWV DQDO\WLFDO UHVXOWV

FRQFHUQLQJ WKH nUPV RSWLPDO LQYHVWPHQW SROLFLHV� WKH YDOXDWLRQ RI WKH 5' YHQWXUH� DQG WKH

UHTXLUHG ULVN SUHPLXP LW HDUQV� 3URSHUWLHV RI WKHVH YDULDEOHV� DQG SDUWLFXODUO\ RI WKH ULVN SUHPLXP�

DUH VWXGLHG QXPHULFDOO\ LQ 6HFWLRQ �� ZKLFK DOVR HYDOXDWHV WKH GLVWULEXWLRQ RI UHDOL]HG UHWXUQV�

6HFWLRQ � FRQFOXGHV� 0RUH WHFKQLFDO GHULYDWLRQV DQG WKH SURRIV WR WKH SURSRVLWLRQV LQ WKH SDSHU DUH

FRQWDLQHG LQ WKH $SSHQGL[�

�

� 5HYLHZ RI 5HODWHG /LWHUDWXUH

$ ODUJH HFRQRPLFV OLWHUDWXUH VWXGLHV G\QDPLF UHVHDUFK DQG GHYHORSPHQW SROLFLHV IRU LQGLYLGXDO

nUPV� �6HH� IRU H[DPSOH� 'XWWD �� *URVVPDQ DQG 6KDSLUR �� 3DNHV �� 3RVQHU DQG

=XFNHUPDQ �� :HLW]PDQ �� RU :HLW]PDQ� 1HZH\ DQG 5DELQ �� $QRWKHU OLQH RI

UHVHDUFK HYDOXDWHV VRFLDOO\ RSWLPDO OHYHOV DQG DOORFDWLRQV RI LQYHVWPHQW LQ UHVHDUFK� DQG WKH HmHFWV

RI SDWHQW ODZ DQG SROLF\� �6HH� IRU H[DPSOH� %KDWWDFKDU\D DQG 0RRNKHUMHH �� 'DVJXSWD DQG

0DVNLQ �� DQG *UXYHU �� 7KH IRFXV RI DOO WKHVH PRGHOV� KRZHYHU� LV RQ RSWLPDO SROLFLHV

UDWKHU WKDQ RQ YDOXDWLRQ� 7KH\ W\SLFDOO\ DVVXPH� IRU H[DPSOH� ULVN QHXWUDOLW\ RU D FRQFDYH XWLOLW\

IXQFWLRQ IRU WKH nUP� 7KLV SUHFOXGHV D VWXG\ RI WKH GHWHUPLQDWHV RI V\VWHPDWLF ULVN DQG ULVN SUHPLD�

DQG WKHLU UROH LQ YDOXDWLRQ� ZKLFK LV ZKDW ZH IRFXV RQ�

0RUH UHFHQWO\� D QXPEHU RI SDSHUV KDYH VWXGLHG WKH 5' SURFHVV DV D FRQWLQJHQW FODLP RQ

WKH YDOXH DQ XQGHUO\LQJ DVVHW� ZKLFK LV LQWHUSUHWHG DV WKH SUHVHQW YDOXH RI WKH FDVK oRZV RQ FRP�

SOHWLRQ RI WKH 5'� 7KLV DSSURDFK DOORZV IRU ULFKHU VSHFLnFDWLRQV WKDQ PRUH WUDGLWLRQDO G\QDPLF

SURJUDPPLQJ VROXWLRQV� DQG LV FORVHU LQ VSLULW WR RXU SDSHU� 7KH YDOXDWLRQ PHWKRG LQ WKHVH SDSHUV

GLmHUV IURP RXUV� KRZHYHU� 7KH\ WDNH DV H[RJHQRXV D VWRFKDVWLF SURFHVV IRU WKH EHQHnW RU YDOXH RI

WKH SURMHFW WR WKH nUP RQ FRPSOHWLRQ RI GHYHORSPHQW� 7KLV HYROYHV DV D GLmXVLRQ� DQG WKH SURMHFW

LV YDOXHG DV D FRQWLQJHQW FODLP RQ WKLV ?XQGHUO\LQJ VHFXULW\�� :H WDNH DV H[RJHQRXV WKH SURFHVV

GHVFULELQJ WKH FDVK oRZV WKH SURMHFW ZLOO JHQHUDWH� DQG YDOXH WKHVH FDVK oRZV DQG WKH 5' SURMHFW

VLPXOWDQHRXVO\� %\ GHULYLQJ WKH YDOXH RI WKH XQGHUO\LQJ FDVK oRZV UDWKHU WKDQ VSHFLI\LQJ LW H[RJH�

QRXVO\� ZH DUH DEOH WR IRFXV RQ WKH UHODWLYH V\VWHPDWLF ULVN RI 5' DQG WKH XQGHUO\LQJ FDVK oRZV

DQG H[SODLQ WKH G\QDPLFV RI WKH ULVN SUHPLXP RI WKH 5'�

0DMG DQG 3LQG\FN �� VROYH DQ LQYHVWPHQW SUREOHP LQ ZKLFK WKH SURMHFW UHTXLUHV D n[HG WRWDO

LQYHVWPHQW WR FRPSOHWH� ZLWK D PD[LPXP LQVWDQWDQHRXV UDWH RI LQYHVWPHQW� $OO WKH XQFHUWDLQW\ LQ

WKHLU PRGHO FRPHV IURP VWRFKDVWLF HYROXWLRQ RI WKH YDOXH RI WKH SURMHFW XSRQ FRPSOHWLRQ� 7KHUH LV

QR WHFKQLFDO ULVN� 7KH SDSHU IRFXVHV RQ KRZ WKH GHFLVLRQ WR LQYHVW RU ZDLW YDULHV ZLWK WKH SDUDPHWHUV

DQG WKH VLQJOH VWDWH YDULDEOH LQ WKH PRGHO�

&KLOGV DQG 7ULDQWLV �� GHYHORS DQG QXPHULFDOO\ LPSOHPHQW D PRGHO RI G\QDPLF 5' WKDW

KLJKOLJKWV WKH LQWHUDFWLRQV DFURVV SURMHFWV� 7KH\ VROYH IRU DQG LQWHUSUHW RSWLPDO SROLFLHV IRU D nUP

ZLWK PXOWLSOH 5' SURMHFWV� ZKLFK FDQ UXQ LQ SDUDOOHO RU VHTXHQWLDOO\� DQG FDOFXODWH WKH YDOXHV

RI WKH UHDO RSWLRQV VXFK SUREOHPV SUHVHQW� 7KHLU PRGHO GLmHUV IURP RXUV ERWK LQ IXQFWLRQ DQG LQ

SXUSRVH� 7KH ULFKQHVV RI WKHLU VSHFLnFDWLRQ DOORZV WKHP WR DQDO\]H LQ GHWDLO WKH LQWHQVLW\ DQG WLPLQJ

RI RSWLPDO LQYHVWPHQW SROLFLHV� 7KH UDQJH RI SROLFLHV ZH DOORZ IRU LV PRUH OLPLWHG� 7KH SUREDOLVWLF

VWUXFWXUHV RI WKH WZR PRGHOV GLmHU� ,QYHVWPHQW GHWHUPLQLVWLFDOO\ \LHOGV SURJUHVV WKURXJK LQYHVWPHQW

VWDJHV LQ WKHLU PRGHO� ZKHUH DV LQ RXU PRGHO VXFFHVV LV UDQGRP� 7KH LQIRUPDWLRQ oRZV LQ WKH WZR

PRGHOV DUH DOVR GLmHUHQW� ,Q &KLOGV DQG 7ULDQWLV �� WKH nUP OHDUQV DERXW WKH XQGHUO\LQJ YDOXH

RQO\ LI LW LQYHVWV� 2WKHUZLVH WKH GLmXVLRQ FRHpFLHQW LQ WKH YDOXH SURFHVV LV VHW HTXDO WR ]HUR� DQG

WKH YDOXH ?KROGV VWLOO�� ,Q RXU PRGHO� LQYHVWLQJ UHVROYHV WHFKQLFDO XQFHUWDLQW\� ,QIRUPDWLRQ DERXW

WKH SRWHQWLDO FDVK oRZV FRQWLQXHV WR HYROYH LQGHSHQGHQWO\ RI WKH nUPV LQYHVWPHQW SROLF\�

�

6FKZDUW] DQG 0RRQ �� DOVR XVH FRQWLQXRXV�WLPH PHWKRGV WR QXPHULFDOO\ YDOXH 5' DV D

FRQWLQJHQW FODLP RQ WKH YDOXH RI WKH EHQHnWV FUHDWHG� 7KHLU SDSHU LV FORVHVW WR RXUV LQ WKH VHQVH

WKDW LW DOORZV VLPXOWDQHRXVO\ IRU WKUHH W\SHV RI XQFHUWDLQW\� 7KHUH LV ?WHFKQLFDO� XQFHUWDLQW\ DVVR�

FLDWHG ZLWK WKH VXFFHVV RI WKH 5' SURFHVV LWVHOI� 7KHUH LV D DQ H[RJHQRXV FKDQFH IRU REVROHVFHQFH�

GXULQJ DQG DIWHU WKH GHYHORSPHQW SURFHVV� DQG WKHUH LV XQFHUWDLQW\ DERXW WKH YDOXH RI WKH SURMHFW

RQ FRPSOHWLRQ RI WKH 5'� 6FKZDUW] DQG 0RRQ �� VROYH IRU RSWLPDO LQYHVWPHQW SROLFLHV� SUR�

YLGH FRPSDUDWLYH VWDWLFV UHJDUGLQJ WKH RSWLRQ FRPSRQHQW RI WKH SURMHFWV YDOXH� DQG FRPSDUH WKH

SURMHFWV YDOXH WR WKH ?139 DSSURDFK� ZKLFK VLPSO\ GLVFRXQWV H[SHFWHG FDVK oRZV JLYHQ RSWLPDO

SROLFLHV� 7KH\ WDNH WKH YDOXH RI WKH SURMHFW RQ FRPSOHWLRQ WR EH H[RJHQRXV� KRZHYHU� DQG GR QRW

HYDOXDWH WKH UHODWLYH V\VWHPDWLF ULVN RI 5' DQG WKH XQGHUO\LQJ FDVK oRZV� 7KH\ DOVR PRGHO

WHFKQLFDO XQFHUWDLQW\ GLmHUHQWO\ WKDQ ZH GR� ,Q WKHLU PRGHO� WKH H[SHFWHG FRVW WR FRPSOHWLRQ RI

WKH SURMHFW LV DQ H[RJHQRXV VWRFKDVWLF SURFHVV ZLWK GULIW DQG GLmXVLRQ FRHpFLHQWV WKDW GHSHQG RQ

LWV FXUUHQW YDOXH DQG RQ WKH FXUUHQW OHYHO RI LQYHVWPHQW� *LYHQ WKLV SURFHVV� RSWLPDO LQYHVWPHQW

SROLFLHV DUH REWDLQHG� DQG JLYHQ WKRVH SROLFLHV WKH H[SHFWHG FRVW WR FRPSOHWLRQ RI WKH SURMHFW PD\

QRW KDYH WKH IXQFWLRQDO IRUP H[RJHQRXVO\ SRVWXODWHG� 2XU PRGHO WDNHV DV H[RJHQRXV WKH WHFKQRO�

RJ\ IRU UDQGRPO\ DGYDQFLQJ WKURXJK VWDJHV RI WKH SURMHFW� DQG WKHQ GHULYHV RSWLPDO LQYHVWPHQW

SROLFLHV� 7KHVH ZLOO OHDG WR DQ HQGRJHQRXV SURFHVV IRU H[SHFWHG FRVW WR FRPSOHWLRQ� ZKLFK ZLOO KDYH

WKH GHVLUDEOH IHDWXUHV RI WKH SURFHVV DVVXPHG E\ 6FKZDUW] DQG 0RRQ �� DV ZHOO DV 3LQG\FN

�� 6SHFLnFDOO\� LQ RXU PRGHO LQQRYDWLRQV LQ WKH FRQGLWLRQDO H[SHFWDWLRQ RI UHPDLQLQJ WLPH WR

FRPSOHWLRQ DUH ]HUR ZKHQ LQYHVWPHQW LV ]HUR� VR WKHUH LV OHDUQLQJ E\ GRLQJ�

� 7KH 0RGHO

�� 7HFKQRORJ\

&RQVLGHU D nUP ZRUNLQJ RQ DQ 5' SURMHFW WR EULQJ D QHZ SURGXFW� SURFHVV RU WHFKQRORJ\ WR

WKH PDUNHW� 6XFFHVVIXO FRPSOHWLRQ RI WKH SURMHFW LQYROYHV FRPSOHWLQJ 1 VWDJHV� ZKHUH 1 LV DQ

H[RJHQRXVO\ JLYHQ LQWHJHU VSHFLnF WR WKH SURMHFW� ,I DQG ZKHQ LW LV FRPSOHWHG� WKH SURMHFW ZLOO

JHQHUDWH D FDVK oRZ VWUHDP� 7KHVH FDVK oRZV DUH VXEMHFW WR WZR W\SHV RI VKRFNV� 7KHUH DUH

SHUVLVWHQW VKRFNV WR WKH OHYHO RI WKH FDVK oRZ HDFK SHULRG� PRGHOHG WKURXJK D UDQGRP ZDON� 7KHUH

LV DOVR D FKDQFH RI D FDWDVWURSKLF HYHQW �OLNH D FRPSHWLWRU GHYHORSLQJ D VXSHULRU SURGXFW� WKDW

H[WLQJXLVKHV WKH FDVK oRZ IURP WKH SURMHFW IRUHYHU� 7KHVH VKRFNV DUH UHDOL]HG ZKHWKHU WKH FDVK

oRZV DUH SRWHQWLDO� EHFDXVH WKH 5' KDV \HW WR EH FRPSOHWHG� RU DFWXDO� EHFDXVH DOO 1 VWDJHV KDYH

EHHQ SDVVHG�

:H PRGHO WKH FDVK oRZV IURP WKH SURMHFW ZLWK WZR VWRFKDVWLF SURFHVVHV Ip�W�� [�W�� W � 7 J� ZKHUH7 GHQRWHV WKH VHW RI SRVLWLYH LQWHJHUV� 7KH SURFHVV I[�W�� W � 7 J WUDFNV ZKHWKHU D FDWDVWURSKLF HYHQW�VXFK DV WKH DSSHDUDQFH RI D SURGXFW VXSHULRU WR WKH RQH EHLQJ GHYHORSHG� KDV RFFXUUHG� ,I [�W� �

IRU DQ\ W � 7 WKHQ WKH FDWDVWURSKLF HYHQW KDV RFFXUUHG� ,I [�W� �� LW KDV QRW� :H UHIHU WR WKH

�

FDWDVWURSKLF HYHQW DV ?REVROHVFHQFH�� $VVXPH WKDW

[�W� �� [�W�\�W� ��

ZKHUH� IRU DOO W � 7 � \�W � �� LV LQGHSHQGHQW RI WKH SDVW DQG LQGHSHQGHQW RI DOO RWKHU UDQGRPYDULDEOHV LQ WKH PRGHO� 7KH YDULDEOH \�W� LV DVVXPHG WDNH YDOXH � �ZKHQ QR FDVWDVWURSKLF HYHQW

RFFXUV� ZLWK SUREDELOLW\ {\ DQG D YDOXH � �ZKHQ D FDWDVWURSKLF HYHQW GRHV RFFXU� ZLWK SUREDELOLW\

�b {\�

7KH SURFHVV Ip�W�� W � 7 J VSHFLnHV WKH FDVK oRZ FRQGLWLRQDO RQ QR REVROHVFHQFH� $W DQ\ GDWH W�ZKLOH WKH 5' LV RQJRLQJ� p�W�[�W� PHDVXUHV WKH FXUUHQW FDVK oRZ WKH SURMHFW ZRXOG KDYH SURGXFHG

LI LW ZHUH FRPSOHWHG� %HIRUH WKH 5' SURMHFW LV FRPSOHWH� WKHUHIRUH� WKH SURFHVV Ip�W�J PRGHOV WKHSHULRGLF DUULYDO RI LQIRUPDWLRQ DERXW WKH SRWHQWLDO FDVK oRZV IURP WKH YHQWXUH� ,I WKH 5' LV

VXFFHVVIXOO\ FRPSOHWHG RQ GDWH W WKHQ IRU DOO ~ w W� p�~�[�~� PHDVXUHV WKH DFWXDO FDVK oRZV IURP

WKH SURMHFW�

:H DVVXPH WKDW p�W� HYROYHV RYHU WLPH DV IROORZV�

p�W� �� p�W� H[S

��xb �

�}�p � � }pqp�W� ��

w��

ZKHUH Iqp�W�� W � 7 J LV DQ LQGHSHQGHQWO\ DQG LGHQWLFDOO\ GLVWULEXWHG FROOHFWLRQ RI VWDQGDUG QRUPDOUDQGRP YDULDEOHV�

$W HDFK GDWH EHIRUH WKH SURMHFW LV FRPSOHWH� WKH nUP PXVW GHFLGH ZKHWKHU WR FRQWLQXH ZRUNLQJ

RQ WKH SURMHFW LQ DQ DWWHPSW WR UHDFK WKH QH[W VWDJH� 7R GR VR� LW PXVW LQFXU D GHYHORSPHQW FRVW

RI ,�W�� 7KLV FDQ EH WKRXJKW RI DV WKH VDODU\ WR EH SDLG WR WKH UHVHDUFK VFLHQWLVWV DQG GHYHORSPHQW

HQJLQHHUV RU WKH FRVW RI UHVHDUFK HTXLSPHQW� GDWD� PDUNHW UHVHDUFK DQG DQDO\VLV� 7KH GHYHORSPHQW

FRVWV FDQ KDYH ERWK D n[HG DQG D UDQGRP� YDULDEOH FRPSRQHQW�

,�W� D� Ep�W�[�W� ��

:H DOORZ WKH SRVVLELOLW\ IRU FRVWV WR KDYH D SURSRUWLRQDO FRPSRQHQW EHFDXVH GHYHORSPHQW FRVWV DUH

OLNHO\ WR EH LQoXHQFHG E\ VXFK IDFWRUV DV LQoDWLRQ DQG ZDJH UDWHV� IRU H[DPSOH� ,W DOVR FDSWXUHV�

WR VRPH H[WHQW� WKH SUDFWLFH RI EDVLQJ VRPH SDUW RI HPSOR\HHV FRPSHQVDWLRQ RQ DQ HTXLW\ VWDNH LQ

WKH RXWFRPH RI WKH 5'� $W WLPHV ZH ZLOO FRQVLGHU VSHFLDO FDVHV ZKHUH D RU E HTXDOV ]HUR WR REWDLQ

WUDFWDELOLW\� RU IRFXV RQ SDUWLFXODU DVSHFWV RI WKH SUREOHP�

�� /HDUQLQJ

,I� DW DQ\ VWDJH Q � 1 WKH nUP PDNHV WKH UHTXLUHG LQYHVWPHQW� WKHQ ZH DVVXPH WKDW ZLWK SUREDELOLW\

{ LW VXFFHHGV DQG PRYHV RQ WR WKH QH[W VWDJH� :LWK SUREDELOLW\ � b {� LW PDNHV QR SURJUHVV DQG

VWD\V ZKHUH LW ZDV� 7KH SDUDPHWHU { LV WKXV D PHDVXUH RI KRZ SURGXFWLYH WKH nUPV 5' DFWLYLWLHV

DUH� :H ZLOO FRQVLGHU FDVHV ZKHUH WKLV LV NQRZQ WR WKH nUP� DQG DOVR FDVHV ZKHUH WKH nUP OHDUQV

�

DERXW { E\ REVHUYLQJ WKH VXFFHVV RU IDLOXUH RI LWV SUHYLRXV HmRUWV�

:KHQ WKH SUREDELOLW\ RI VXFFHVV LV QRW NQRZQ� WKH VXFFHVV RU IDLOXUH RI HDFK HmRUW WKH nUP PDNHV

RQ WKH SURMHFW FRQYH\V LQIRUPDWLRQ DERXW WKH XQNQRZQ SUREDELOLW\ RI VXFFHVV� 7R PRGHO WKLV� ZH

DVVXPH WKDW WKH LQLWLDO SULRU GLVWULEXWLRQ RI WKH SUREDELOLW\ RI VXFFHVV DW HDFK VWDJH� {� LV D EHWD

GLVWULEXWLRQ ZLWK SDUDPHWHUV �m�� m�� 7KH GLVWULEXWLRQ RI {� WKH SUREDELOLW\ RI VXFFHVV� LV XSGDWHG

DFFRUGLQJ WR %D\HV UXOH XVLQJ WKH KLVWRU\ RI SDVW DWWHPSWV WR FRPSOHWH DOO WKH YDULRXV VWDJHV RI

UHVHDUFK� ,I� DW D SDUWLFXODU GDWH W� WKH nUP KDV EHHQ VXFFHVVIXO LQ Q DWWHPSWV RXW RI P� Q WULDOV�

WKH nUPV SRVWHULRU GLVWULEXWLRQ RI { RQ GDWH W LV EHWD ZLWK SDUDPHWHUV �m��Q� m��P�� ,I WKH nUP

FKRRVHV WR FRQWLQXH WKH UHVHDUFK HmRUW DW W� DW W� � LW ZLOO nQG RXW LI LW HLWKHU VXFFHHGHG RU IDLOHG�

,I WKH nUP VXFFHHGV� %D\HV UXOH LPSOLHV WKDW WKH SRVWHULRU GLVWULEXWLRQ RI { DW GDWH W � � ZLOO EH

EHWD ZLWK SDUDPHWHUV �m�� Q�� m��P�� ,I WKH nUP IDLOV� WKH SRVWHULRU GLVWULEXWLRQ RI { DW W��

LV EHWD ZLWK SDUDPHWHUV �m� � Q� m� �P� ��

�� 9DOXDWLRQ 3URFHVVHV

7R DQDO\]H WKH YDOXH RI DQ DVVHW ZKRVH FDVK oRZV DUH XQFHUWDLQ� DQG XQGHUVWDQG LWV GHSHQGHQFH

RQ LWV V\VWHPDWLF ULVN� ZH PXVW SXW VWUXFWXUH RQ WKH SULFLQJ NHUQHO LQ WKH HFRQRP\� /HW WKH SULFLQJ

NHUQHO EH GHQRWHG I]�W�� W � 7 J� 7KDW LV WR VD\� WKH GDWH W YDOXH RI DQ\ FDVK oRZ DUULYLQJ DW 7 LVFRPSXWHG DV

(W

�]�7 �

]�W�&�7 �

w� ��

DQG FRYDULDQFH ZLWK WKH SULFLQJ NHUQHO FDQ EH LQWHUSUHWHG DV V\VWHPDWLF ULVN� FRUUHVSRQGLQJ� IRU

H[DPSOH� ZLWK ?EHWD� XQGHU WKH PHDQ�YDULDQFH &$30�

:H DVVXPH WKDW WKH SULFLQJ NHUQHO HYROYHV WKURXJK WLPH DV IROORZV�

]�W� �� ]�W� H[S

�bU b �

�}�] b }]q]�W� ��

w��

ZKHUH U GHQRWHV WKH RQH�SHULRG ULVNOHVV UDWH� DQG Iq]�W�� W � 7 J LV DQ LQGHSHQGHQWO\ DQG LGHQWLFDOO\GLVWULEXWHG FROOHFWLRQ RI UDQGRP YDULDEOHV� :H DVVXPH WKH ULVNOHVV UDWH LV FRQVWDQW� 7KLV LV GRQH WR

IRFXV RQ ULVN SUHPLD WKDW DUH DWWULEXWDEOH WR V\VWHPDWLF ULVN WKDW� LQ RXU VHWWLQJ� LV DQDORJRXV WR

?PDUNHW ULVN� UDWKHU WKDQ LQWHUHVW UDWH ULVN�

)LQDOO\� ZH DVVXPH WKDW IRU DQ\ W� WKH SDLU RI LQQRYDWLRQV� Iqp�W�� q]�W�J� KDV D ELYDULDWH QRUPDOGLVWULEXWLRQ� ZKHUH WKH PHDQV RI ERWK YDULDEOHV DUH ]HUR� WKHLU YDULDQFHV DUH RQH� DQG WKHLU FRYDULDQFH

LV |� :H GHQRWH WKH PDUNHW SULFH RI ULVN DV w s |}p}]� 7KLV SDUDPHWHU GHWHUPLQHV WKH HTXLOLEULXP

H[FHVV UHWXUQ HDUQHG E\ KROGLQJ DQ HTXLW\ SRVLWLRQ LQ WKH XQGHUO\LQJ FDVK oRZ SURFHVV� p�W�� ,W WKXV

SURYLGHV D EHQFKPDUN DJDLQVW ZKLFK WR HYDOXDWH WKH ULVN SUHPLD HDUQHG E\ KROGLQJ DQ HTXLW\ VWDNH

�7KHVH FRQFOXVLRQV DUH EDVHG RQ WKH IROORZLQJ IDFW� VXSSRVH� ;L� L �� Q LV D UDQGRP VDPSOH IURP D %HUQRXOOL�ELQRPLDO� GLVWULEXWLRQ ZLWK DQ XQNQRZQ SDUDPHWHU YDOXH : � 6XSSRVH WKDW WKH SULRU GLVWULEXWLRQ RI : LV D EHWDGLVWULEXWLRQ ZLWK SDUDPHWHUV m� ! � DQG m� ! �� 7KHQ� WKH SRVWHULRU GLVWULEXWLRQ RI : � ZKHQ ;L [L� L �� Q�LV D EHWD GLVWULEXWLRQ ZLWK SDUDPHWHUV m� � \ DQG m� � Q b \ ZKHUH \

3Q

L �[L� 6HH 7KHRUHP � LQ 6HFWLRQ �� RI

'H*URRW ��

�

LQ WKH 5' SURMHFW�

,W ZRXOG EH VWUDLJKWIRUZDUG� DOWKRXJK DOJHEUDLFDOO\ DQG FRPSXWDWLRQDOO\ EXUGHQVRPH� WR JHQ�

HUDOL]H WKH PRGHO WR DOORZ IRU VWRFKDVWLF LQWHUHVW UDWHV WKDW IROORZ D VLPSOH RQH�IDFWRU PRGHO� DV LQ

%HUN� *UHHQ DQG 1DLN �� $VVXPLQJ LQWHUHVW UDWHV DUH FRQVWDQW DQG LQQRYDWLRQV WR WKH SULFLQJ

RSHUDWRU DUH LQGHSHQGHQW DQG LGHQWLFDOO\ GLVWULEXWHG WKURXJK WLPH KDV VRPH H[SRVLWLRQDO DGYDQWDJHV

IRU RXU SUREOHP� ,W HQVXUHV WKDW ULVN SUHPLD DUH FRQVWDQW IRU WKH XQGHUO\LQJ FDVK oRZ SURFHVV� DQG

WKDW DOO RI WKH VWDWH YDULDEOHV ZH PXVW WUDFN LQ YDOXLQJ WKH nUP DQG GHWHUPLQLQJ RSWLPDO SROLFLHV

DUH nUP VSHFLnF� 7KXV� DQ\ YDULDELOLW\ LQ WKH ULVN SUHPLD FDQ EH DWWULEXWHG WR WKH VSHFLDO IHDWXUHV RI

WKH 5' YHQWXUH UDWKHU WKDQ PRYHPHQWV LQ WKH HFRQRP\ LQ JHQHUDO� ,Q DGGLWLRQ� WKHVH DVVXPSWLRQV

DOORZ IRU VLPSOH� SDUDPHWULF GHFRPSRVLWLRQV EHWZHHQ V\VWHPDWLF DQG XQV\VWHPDWLF ULVN� ,Q WKLV ZD\

WKH PRGHO FDQ SURYLGH TXDOLWDWLYH JXLGDQFH DERXW KRZ WKH FKDUDFWHULVWLFV RI 5' SURMHFWV VKRXOG

LQoXHQFH MXGJPHQWV DERXW UHTXLUHG UDWHV RI UHWXUQ�

� 9DOXDWLRQ

:H EHJLQ E\ EULHo\ VXPPDUL]LQJ WKH nUPV GHFLVLRQ SUREOHP� $W DQ\ GDWH W� WKH nUP LQ RXU PRGHO

PXVW GHFLGH ZKHWKHU WR LQYHVW LQ DGGLWLRQDO 5' RU QRW� 7KH nUP VROYHV WKLV GLVFUHWH FKRLFH

SUREOHP WR PD[LPL]H LWV FXUUHQW PDUNHW YDOXH� 7KH FRVW RI FRQWLQXLQJ LV WKH DGGLWLRQDO LQYHVWPHQW

UHTXLUHG IRU WKH FXUUHQW SHULRG� ,�W�� 7KH EHQHnW LV WKH FKDQFH WR PRYH DKHDG WKURXJK RQH RI WKH

1 VWDJHV UHTXLUHG WR FRPSOHWH WKH SURMHFW� DQG WKXV JDLQ FRQWURO RI WKH SHULRGLF FDVK oRZV JLYHQ

E\ WKH SURFHVV Ip�W�� W � 7 J� 7KH SUREDELOLW\ WKDW WKH nUP ZLOO PRYH DKHDG D VWDJH� LI LW PDNHV WKHLQYHVWPHQW� LV JLYHQ E\ {� D SDUDPHWHU WKH nUP HYDOXDWHV WKURXJK %D\HV UXOH RQ WKH EDVLV RI SDVW

VXFFHVVHV DQG IDLOXUHV RI LWV 5' HmRUWV� ,I WKH nUP GRHV QRW LQYHVW� LWV LQIRUPDWLRQ VHW DQG YDOXH

ZLOO UHPDLQ n[HG H[FHSW IRU WKH H[RJHQRXV HYROXWLRQ RI p�W��

7KXV� WKH VWDWH YDULDEOHV RQ ZKLFK WKH nUP PXVW EDVH LWV GHFLVLRQ DW W DUH DV IROORZV�

P�W� 7KH QXPEHU RI SUHYLRXV LQYHVWPHQWV WKDW KDYH IDLOHG WR PRYH WKH YHQWXUH WKURXJK D VWDJH RI

WKH 5' SURFHVV�

Q�W� 7KH QXPEHU RI VWDJHV RI WKH 5' SURFHVV WKH YHQWXUH KDV VXFFHVVIXOO\ SDVVHG WKURXJK� 7KH

QXPEHU RI UHPDLQLQJ VWDJHV LV 1 b Q�W��

p�W� 7KH FXUUHQW RU SRWHQWLDO FDVK oRZ IURP WKH SURMHFW�

[�W� $Q LQGLFDWRU HTXDO WR ]HUR ZKHQ WKH SURMHFW LV REVROHWH� DQG RWKHUZLVH RQH�

/HW 9 >Q�W��P�W�� p�W�@ EH WKH FXUUHQW YDOXH RI WKH nUP FRQGLWLRQDO RQ [�W� �� ,I [�W� ��

WKH YDOXH RI WKH YHQWXUH LV ]HUR� ,W LV DOVR DSSDUHQW WKDW LI Q�W� 1 � WKH SURMHFW LV FRPSOHWH DQG

9 �1�P� p� ZLOO MXVW EH WKH SUHVHQW YDOXH RI IXWXUH FDVK oRZV UHJDUGOHVV RI P�

�

�� 9DOXDWLRQ RI WKH &DVK )ORZ 3URFHVV

:H FDQ FRPSXWH WKH YDOXH RI WKH FDVK oRZ SURFHVV DW DQ\ GDWH DV

(W

� �;V �

]�W� V�

]�W�p�W� V�[�W� V�

��

1RWH WKLV LV WKH ?FXP�GLYLGHQG� YDOXH� DV LW LQFOXGHV WKH FXUUHQW FDVK oRZ� 7KLV ZLOO FRLQFLGH ZLWK

WKH YDOXH RI WKH nUP RU YHQWXUH ZKHQ WKH 5' LV FRPSOHWH� 2XU GLVWULEXWLRQDO DVVXPSWLRQV JLYH

WKH YDOXH RI WKH FDVK oRZV D YHU\ VLPSOH IRUP�

3URSRVLWLRQ � 7KH YDOXH RI WKH FRPSOHWHG SURMHFW LV

9 >1�P�W�� p�W�@ p�W�

�b {\Hxb�U�w��

7KLV H[SUHVVLRQ LV UHFRJQL]DEOH DV WKH FRQWLQXRXVO\ FRPSRXQGHG YHUVLRQ RI WKH JURZLQJ SHUSH�

WXLW\ IRUPXOD� +HUH WKH GLVFRXQW UDWH LV WKH VXP RI WKH ULVNOHVV UDWH DQG WKH ULVN SUHPLXP� 7KH IDFW

WKDW WKH FDVK oRZV WHUPLQDWH VWRFKDVWLFDOO\ LV FDSWXUHG E\ {\ LQ WKH GHQRPLQDWRU� )RU FRQYHQLHQFH

ZH ZLOO KHQFHIRUWK GHnQH WKLV GLVFRXQW IDFWRU WR EH

n s {\ H[S>xb �U � w�@� ��

7KH UROH RI V\VWHPDWLF ULVN LQ UHGXFLQJ WKH YDOXH RI WKH FDVK oRZV LV HYLGHQW LQ WKH SUHVHQFH RI

w LQ WKH H[SUHVVLRQ IRU n� ZKHUH LW DXJPHQWV WKH ULVNOHVV UDWH LQ GLVFRXQWLQJ WKH FDVK oRZV� )RU

WKH YDOXH WR EH nQLWH� LW LV REYLRXV WKDW n PXVW EH D IUDFWLRQ� ,Q RXU H[SUHVVLRQ IRU WKH SULFLQJ

NHUQHO� �� WKH LQQRYDWLRQV HQWHU QHJDWLYHO\� VR WKDW KLJKHU w FRUUHVSRQGV WR PRUH �XQGHVLUDEOH�

V\VWHPDWLF ULVN� 7KH IDFW WKDW WKH V\VWHPDWLF ULVN HQWHUV WKH YDOXDWLRQ WKURXJK D PXOWLSOLFDWLYH

FRQVWDQW HQVXUHV WKDW WKH H[SHFWHG UHWXUQ WR KROGLQJ DQ HTXLW\ SRVLWLRQ LQ WKH FRPSOHWHG SURMHFW

LV FRQVWDQW�

3URSRVLWLRQ � 7KH SHU�GROODU FRQWLQXRXVO\ FRPSRXQGHG H[SHFWHG UHWXUQ RQ D VHFXULW\ WKDW SD\V

WKH VWUHDP Ip�W�� W � 7 J LV FRQVWDQW DQG HTXDO WR U � w�

�� 7KH 9DOXH )XQFWLRQ

,Q WKH DSSHQGL[ ZH FRQVWUXFW WKH YDOXH IXQFWLRQ IRU WKH nUPV SUREOHP RI GHFLGLQJ ZKHWKHU WR LQYHVW

LQ 5'� 7KLV IXQFWLRQ VHUYHV ERWK DV WKH REMHFWLYH RI WKH nUPV G\QDPLF SURJUDPPLQJ SUREOHP�

DQG DV WKH PDUNHW YDOXH RI WKH 5' SURMHFW�

�

7R FRQVWUXFW WKLV IXQFWLRQ QRWH WKDW WKH YDOXH WRGD\ LV MXVW WKH H[SHFWHG �GLVFRXQWHG� YDOXH RI

WKH IXQFWLRQ LQ RQH SHULRG� 7KLV H[SHFWDWLRQ PXVW EH WDNHQ RYHU DOO SRVVLEOH VWDWHV� ,I WKH nUP

PDNHV WKH LQYHVWPHQW� WKHVH ZLOO EH WKH GLVFUHWH RXWFRPHV DVVRFLDWHG ZLWK REVROHVFHQFH RU WKH ODFN

RI LW� DQG IDLOXUH RU VXFFHVV LQ PRYLQJ WKURXJK WKH FXUUHQW VWDJH� ,Q DGGLWLRQ WKHUH DUH WKH LQnQLWHO\

PDQ\ VWDWHV FRUUHVSRQGLQJ WR WKH GLmHUHQW SRVVLEOH YDOXHV IRU { DQG p�W� �� ,I WKH nUP GRHV QRW

LQYHVW� WKHQ WKH H[SHFWDWLRQ ZLOO RQO\ EH RYHU WKH VWDWHV FRUUHVSRQGLQJ WR SRVVLEOH REVROHVFHQFH DQG

p�W� �� 7KH GHFLVLRQ RQ ZKHWKHU RU QRW WR PRWKEDOO WKH SURMHFW LV PDGH WR PD[LPL]H YDOXH� WKXV�

IRU Q � 1 � WKH YDOXH IXQFWLRQ PXVW VDWLVI\ WKH IROORZLQJ UHFXUVLRQ�

9 �Q�P� p� ��

PD[

|= �

�>{9H�Q� ��P� p� � ��b {�9H�Q�P� �� p�@ Y�{MP�Q�G{ b Db Ep� 9H�Q�P� p�

}

ZKHUH Y��MQ�P� GHQRWHV WKH FRQGLWLRQDO GLVWULEXWLRQ RI { JLYHQ WKDW WKH nUP KDV PDGH Q VXFFHVVIXODQG P XQVXFFHVVIXO DWWHPSWV WKXV IDU� DQG 9H�Q�P� p� LV WKH FXUUHQW GLVFRXQWHG YDOXH RI WKH nUP

QH[W SHULRG FRQGLWLRQDO RQ� �� EHLQJ FXUUHQWO\ DOLYH DQG �� KROGLQJ n[HG LWV SURJUHVV LQ 5' DW

Q VWDJHV FRPSOHWH RYHU Q�P WULDOV� 7KDW LV�

9H�Q�P� p� ��

(W

|]�W� ��

]�W�[�W� ��9 >Q�P� p�W� ��@ M p�W� p� [�W� �� Q�W� Q�P�W� P

}�

1RWH WKDW WKH PD[Ic� cJ RSHUDWRU LQ �� LPSOHPHQWV WKH GHFLVLRQ RQ ZKHWKHU WR LQYHVW LQ WKH FXUUHQWSHULRG� ,I WKH LQYHVWPHQW LV PDGH� WKH H[SHFWHG YDOXH RI WKH SURMHFW LQ RQH SHULRG UHoHFWV ERWK

ZKHWKHU WKH QH[W VWDJH LV VXFFHVVIXOO\ FRPSOHWHG DQG WKH XSGDWLQJ WKDW RFFXUV WR WKH SUREDELOLW\

RI VXFFHVV� ,I WKH SURMHFW LV PRWKEDOOHG� WKHQ LWV H[SHFWHG YDOXH GRHV QRW GHSHQG RQ HLWKHU�

,Q WKH DSSHQGL[� ZH nUVW VKRZ WKDW WKH UHFXUVLRQ �� KDV D XQLTXH VROXWLRQ� :H WKHQ FRQVLGHU

WKH PRGHO ZLWK D nQLWH�WLPH KRUL]RQ IRU FRPSOHWLRQ RI WKH 5'� 7KDW LV� LI WKH 1 VWDJHV DUH QRW

nQLVKHG E\ VRPH GDWH 7 � WKH SURMHFW EHFRPHV ZRUWKOHVV� :H YHULI\ WKDW WKH VROXWLRQ WR WKH UHFXUVLRQ

LQ �� LV WKH OLPLW RI WKH YDOXH RI WKH 5' SURMHFW DV 7 � �� DQG WKXV WKDW LW LV� LQGHHG� WKH YDOXHRI WKH SURMHFW IRU D nUP IDFLQJ DQ LQnQLWH KRUL]RQ�

6LQFH WKH LQWHJUDQG LQVLGH WKH PD[Ic� cJ RSHUDWRU RQ WKH ULJKW�KDQG VLGH RI �� LV OLQHDU LQ {� WKH

DERYH UHFXUVLRQ FDQ EH UHVWDWHG DV�

9 �Q�P� p� ��

PD[ Ix{�Q�P�9H�Q� ��P� p� � ��b x{�Q�P��9H�Q�P� �� p�b Db Ep� 9H�Q�P� p�J

ZKHUH

x{�Q�P�

= �

�{Y�{MQ�P�G{ m� � Q

m� � m� � Q�P� ��

LV WKH PHDQ RI WKH nUPV SRVWHULRU GLVWULEXWLRQ RYHU {�

��

:H QH[W HVWDEOLVK D QXPEHU RI JHQHUDO SURSHUWLHV WKH YDOXH RI WKH 5' YHQWXUH PXVW VDWLVI\�

3URSRVLWLRQ � 7KH YDOXH RI WKH 5' YHQWXUH� 9 �Q�P� p��

�� LV PRQRWRQH DQG FRQYH[ LQ p� WKH YDOXH RI WKH FDVK oRZV

�� LV LQFUHDVLQJ LQ Q� WKH QXPEHU RI VWDJHV FRPSOHWHG� DQG GHFUHDVLQJ LQ P� WKH QXPEHU RI IDLOXUHV

�� DSSURDFKHV ]HUR DV p� WKH SRWHQWLDO FDVK oRZ� DSSURDFKHV ]HUR

7KLV SURSRVLWLRQ VKRZV WKDW WKH YDOXH RI WKH YHQWXUH UHVSRQGV LQ SUHGLFWDEOH ZD\V WR WKH VWDWH

YDULDEOHV p� Q DQG P� 6XFFHVVIXOO\ SDVVLQJ WKURXJK 5' VWDJHV LV JRRG QHZV� ,QWXLWLYHO\� D KLJKHU

Q RQO\ UHOD[HV FRQVWUDLQWV RQ WKH nUP LQ LWV JRDO RI UHDOL]LQJ FDVK oRZV� 6LPLODUO\� KLJKHU IDLOXUH

UDWHV DUH EDG QHZV� 7KH nUPV SRVWHULRU DVVHVVPHQW RI LWV FKDQFHV RI HYHU UHDOL]LQJ WKH FDVK oRZV

ZLOO GHFUHDVH DV P LQFUHDVHV� KROGLQJ Q n[HG� 7KH PRQRWRQLFLW\ DQG FRQYH[LW\ LQ p IROORZV GLUHFWO\

IURP WKH YDOXH IXQFWLRQV RSWLRQ�OLNH SURSHUWLHV� 6LQFH WKH YDOXH RI WKH nUP LV DOZD\V WKH H[SHFWHG

GLVFRXQWHG YDOXH RI LWV IXWXUH FDVK oRZV� LWV YDOXH PXVW DSSURDFK ]HUR DV WKHVH FDVK oRZV DSSURDFK

]HUR�

�� 3URSRUWLRQDO 5' &RVWV

7KH FDVH ZLWK QR n[HG FRVWV �D �� LV LQWHUHVWLQJ IRU VHYHUDO UHDVRQV� $ FORVHG�IRUP �XS WR D nQLWH

VXPPDWLRQ� VROXWLRQ IRU WKH YDOXH LV DWWDLQDEOH VR WKDW SURSHUWLHV RI WKH RSWLPDO LQYHVWPHQW SROLF\

FDQ EH HDVLO\ HVWDEOLVKHG DQG LQWXLWLYHO\ HYDOXDWHG� 7KLV LQYHVWPHQW SROLF\ KDV D YHU\ VLPSOH IRUP�

7KH nUP ZLOO QHYHU ?PRWKEDOO� WKH SURMHFW LQ KRSHV WKDW FRQGLWLRQV ZLOO LPSURYH DQG OHDG WR D

UHVXPSWLRQ RI GHYHORSPHQW� ,W ZLOO RQO\ DEDQGRQ WKH SURMHFW� 7KLV DEDQGRQPHQW GHFLVLRQ LV GULYHQ

HQWLUHO\ E\ OHDUQLQJ DERXW WKH nUPV UHVHDUFK SURGXFWLYLW\ �{�� 7KHVH UHVXOWV ZLOO SURYH LPSRUWDQW

ZKHQ ZH UHWXUQ WR WKH JHQHUDO FDVH LQ 6HFWLRQ �� 7KH VROXWLRQ LQ WKH SURSRUWLRQDO FRVW FDVH LV DOVR

UHTXLUHG WR HVWDEOLVK VRPH RI WKH SURSHUWLHV RI WKH JHQHUDO VROXWLRQ� ,W ZLOO DOVR EH XVHG LQIRUPDOO\

DV D EHQFKPDUN WR HYDOXDWH WKH LQFUHPHQWDO HmHFWV RI WKH GHFLVLRQ WR PRWKEDOO�

7KH WUDFWDELOLW\ RI WKLV FDVH GHULYHV IURP WKH IDFW WKDW WKH YDOXH IXQFWLRQ LV SURSRUWLRQDO WR WKH

OHYHO RI WKH FDVK oRZ SURFHVV�

3URSRVLWLRQ � ,I ,�W� Ep�W� WKHQ WKH YDOXH IXQFWLRQ LV KRPRJHQHRXV RI GHJUHH RQH LQ p� DQG ZH

KDYH

9 �Q�P� p� pK�Q�P��

DQG

9H�Q�P� p� pnK�Q�P��

ZLWK n {\ H[S>xb �U � w�@� 7KH IXQFWLRQ K�Q�P� VDWLVnHV WKH UHFXUVLRQ�

K�Q�P� PD[ >x{�Q�P�nK�Q� ��P� � ��b x{�Q�P��nK�Q�P� ��b E� nK�Q�P�@ ��

��

7R REWDLQ DQG H[SUHVVLRQ IRU K�Q�P�� ZH PXVW nUVW GHULYH DSSURSULDWH ERXQGDU\ FRQGLWLRQV�

7KHQ ZH ZLOO VKRZ WKDW WKH RSWLPDO LQYHVWPHQW VWUDWHJ\ UHVSRQGV WR H[SHULHQFH LQ D XQLIRUP ZD\

_ WKH nUP RQO\ DEDQGRQV ZKHQ WKH QXPEHU RI IDLOXUHV UHDFKHV VRPH WKUHVKROG ZKLFK ZH FDOO Pe�Q��7KLV DOORZV XV WR VXEVWLWXWH WKLV XSSHU ERXQG RQ SDVW IDLOXUHV IRU WKH PD[>c� c@ RSHUDWRU� DQG GHULYHD VROXWLRQ WKURXJK UHFXUVLYH VXEVWLWXWLRQ� 7KH IROORZLQJ SURSRVLWLRQV DFFRPSOLVK WKHVH WDVNV�

3URSRVLWLRQ � :KHQ ,�W� Ep�W�� D �� E ! �� PRWK�EDOOLQJ LV HTXLYDOHQW WR DEDQGRQPHQW� DQG

LI K�Q�P� VROYHV �� LW DOVR VROYHV

K�Q�P� PD[ >x{�Q�P�nK�Q� ��P� � ��b x{�Q�P��nK�Q�P� ��b E� �@ � ��

)XUWKHUPRUH� K�1�P� ��bn IRU DOO P� DQG IRU HYHU\ Q � 1 WKHUH PXVW H[LVW DQ LQWHJHU Pe�Q� � �

VXFK WKDW IRU DOO P ! Pe�Q�� K�Q�P� ��

7KH nQLWHQHVV RI Pe�Q� DQG WKH YDOXH RI WKH 5' XSRQ FRPSOHWLRQ SURYLGH WKH QHFHVVDU\

ERXQGDU\ FRQGLWLRQV� %\ WKH GHnQLWLRQ RI Pe�Q�� K�Q�P� ! � IRU DOO P � Pe�Q�� 6R IRU DOOP � Pe�Q��

K�Q�P� x{�Q�P�nK�Q� ��P� � ��b x{�Q�P��nK�Q�P� ��b E ��

DQG

K�Q�Pe�Q�� x{�Q�Pe�Q��nK�Q� ��Pe�Q��b E� ��

7KXV� Pe�Q� FDQ EH FRPSXWHG DV

PD[IPMx{�Q�P�nK�Q� ��P�b E ! �J ��

$ FORVHG�IRUP H[SUHVVLRQ IRU WKH IXQFWLRQV K�Q�P� DUH FRPSXWHG UHFXUVLYHO\ DV IROORZV� 6XSSRVH

WKDW ZH DOUHDG\ KDYH WKH VROXWLRQ WR K�Q�� c�� 2QFH ZH KDYH Pe�Q�� ZH FDQ FRPSXWH WKH YDOXH RIK�Q�P� IRU P � Pe�Q�� VLPSO\ E\ UHFXUVLYH VXEVWLWXWLRQ LQ �� 7KH IROORZLQJ SURSRVLWLRQ VWDWHVWKH UHVXOWV RI WKLV SURFHGXUH�

3URSRVLWLRQ �

K�Q�P�

�!!!!�!!!!�

��bn � LI Q 1

�� LI Q � 1 DQG P ! Pe�Q�3Pe�Q�bPL �

Knr

Q�m�P�Q�L�m��m�

sK�Q� ��P� L�b E

LnL

bP�m�

cLb

P�Q�m��m�cL

� R�Z�

��

ZKHUH Pe�Q� LV GHnQHG LPSOLFLWO\ DV WKH ODUJHVW P VXFK WKDW

P ��Q� m��n K�Q� ��P�

Eb Qb m� b m��

��

DQG �m�Q s b�m�Q�b�m� LV WKH 3RFKKDPPHU V\PERO�

7KH QH[W VWHS LV WR XVH WKH RWKHU ERXQGDU\ FRQGLWLRQ WKDW K�1�P� � ��bn� LQ �� WR SURYLGH

D FORVHG�IRUP H[SUHVVLRQ IRU K�1 b �� c�� 7KLV H[SUHVVLRQ FDQ WKHQ EH XVHG LQ �� WR UHFXUVLYHO\REWDLQ WKH FRPSDUDEOH FORVHG�IRUP H[SUHVVLRQ IRU DOO Q � 1 b ��:KHQ { LV NQRZQ ZH FDQ GURS P� ZKLFK VXPPDUL]HV WKH KLVWRU\ SDVW LQYHVWPHQW LQ WKH SURMHFW�

DV D VWDWH YDULDEOH� 'HnQH

J�Q� s K�Q�P��

VR WKDW WKH YDOXH RI WKH SURMHFW LV J�Q�p� ,I J�� WKH SURMHFW LV QHYHU VWDUWHG VR J�Q� �� DQG

ZH FDQ OLPLW DWWHQWLRQ WR WKH FDVH ZKHQ J�� ! �� 3URSRVLWLRQ �� VKRZV WKDW WKH YDOXH LV LQFUHDVLQJ

LQ Q� WKH QXPEHU RI VWDJHV SDVVHG WKURXJK� 7KXV� LQ WKLV FDVH WKH SURMHFW ZLOO QHYHU EH DEDQGRQHG�

6XEVWLWXWLQJ �� LQWR �� SURYLGHV�

J�Q� {nJ�Q� �� b {�nJ�Q�b E ��

IRU Q �� 1 b �� %\ 3URSRVLWLRQ � LW PXVW DOVR EH WKH FDVH WKDW

J�1� K�1�P� �

�b n� ��

7KH IROORZLQJ SURSRVLWLRQ SURYLGHV WKH VROXWLRQ WR WKLV UHFXUVLRQ�

3URSRVLWLRQ � :KHQ ,�W� Ep�W� DQG { LV NQRZQ� WKH YDOXH RI WKH 5' YHQWXUH DW GDWH W JLYHQ

WKDW LW KDV DWWDLQHG VWDJH Q LV p�W�J�Q� ZKHUH LI

J�� s��

�b n�

Et

�b o

wo1 b Et

�b o! � ��

ZLWK

o s {n

�b ��b {�n��

DQG

t s �

�b ��b {�n��

WKHQ

J�Q�

��

�b n�

Et

�b o

wo1bQ b Et

�b o��

DQG RWKHUZLVH J�Q� � IRU DOO Q � 1 � )XUWKHUPRUH� WKH YDOXH RI WKH SURMHFW LV LQFUHDVLQJ DQG

FRQYH[ LQ QXPEHU RI VWDJHV FRPSOHWHG�

�� 2SWLPDO ,QYHVWPHQW 3ROLF\

7KH SURSHUWLHV RI WKH VROXWLRQ XQGHU SURSRUWLRQDO FRVWV DUH XVHIXO IRU XQGHUVWDQGLQJ WKH VROXWLRQ LQ

WKH JHQHUDO FDVH� )RU H[DPSOH� LW LV HYLGHQW WKDW WKH VROXWLRQV IRU WKH JHQHUDO DQG WKH SURSRUWLRQDO

��

FDVH FRQYHUJH DV WKH YDOXH RI WKH FDVK oRZ JURZV DQG WKH ?OHYHUDJH� DVVRFLDWHG ZLWK WKH n[HG FRVWV

EHFRPHV OHVV LPSRUWDQW� 7KH QH[W SURSRVLWLRQ SURYHV WKLV�

3URSRVLWLRQ � $V p�W� JURZV ZLWKRXW ERXQG� WKH YDOXH RI WKH SURMHFW DSSURDFKHV LWV YDOXH IRU WKH

SURSRUWLRQDO FDVH�

OLPp��

9 �Q�P� p� ��

p K�Q�P� ��

7KH RSWLPDO SROLFLHV IRU WKH SURSRUWLRQDO FDVH DOVR KHOS WR GHOLQHDWH WUDGHRmV WKDW JRYHUQ RSWLPDO

SROLFLHV LQ WKH JHQHUDO FDVH� :KHQ FRVWV DUH VWULFWO\ SURSRUWLRQDO� ZKLOH WKH RSWLRQ WR VXVSHQG 5'

LV YDOXDEOH� RQFH VXVSHQGHG WKH SURMHFW LV QHYHU UHVWDUWHG� ,QWXLWLYHO\� WKLV LV EHFDXVH QR OHDUQLQJ

DERXW WKH nUPV 5' FDSDELOLWLHV �{� WDNHV SODFH ZLWKRXW LQYHVWPHQW� /HDUQLQJ DERXW WKH nUPV

FDVK oRZV� E\ REVHUYLQJ p�W�� GRHV QRW OHDG WR D UHVXPSWLRQ RI 5' VSHQGLQJ� EHFDXVH LW HmHFWV

ERWK FRVWV DQG EHQHnWV SURSRUWLRQDOO\�

7KH RWKHU H[WUHPH LV ZKHQ FRVWV KDYH QR SURSRUWLRQDO FRPSRQHQW� ,Q WKLV FDVH� WKHUH LV QR UHDVRQ

WR HYHU DEDQGRQ WKH SURMHFW� 2QFH LQYHVWPHQW LV VXVSHQGHG� OHDUQLQJ DERXW WKH nUPV FDSDELOLWLHV

FHDVHV� DQG OHDUQLQJ DERXW WKH YDOXH RI WKH FDVK oRZV FRQWLQXHV� 7KHUH LV DOZD\V D FKDQFH WKDW p�W�

ZLOO JURZ VXpFLHQWO\ ODUJH WKDW LW ZLOO UHQGHU WKH n[HG FRVWV LQVLJQLnFDQW UHODWLYH WR WKH SURMHFWV

SRWHQWLDO YDOXH�

3URSRVLWLRQ � :KHQ ,�W� D� �E �� WKH nUP DOZD\V PRWKEDOOV DQG QHYHU DEDQGRQV WKH 5'

SURMHFW� ZKLOH ZKHQ ,�W� Ep�W�� D �� WKH nUP DOZD\V DEDQGRQV DQG QHYHU PRWKEDOOV WKH SURMHFW�

2XU QH[W UHVXOW HVWDEOLVKHV WKH RSWLPDO LQYHVWPHQW SROLF\ IRU WKH JHQHUDO FDVH� :H nUVW HVWDEOLVK

WKDW WKH nUP ZLOO HYHQWXDOO\ JLYH XS LQ WKH IDFH RI UHSHDWHG IDLOXUH DQG H[HUFLVH LWV DEDQGRQPHQW

RSWLRQ� 7R VHH WKLV� QRWH WKDW IRU n[HG E� WKH YDOXH RI WKH SURMHFW ZLWK D � PXVW EH JUHDWHU WKDQ

ZLWK D ! �� 7KXV� VLQFH 3URSRVLWLRQ � VKRZV WKDW DQ P H[LVWV ZKHQ LW LV RSWLPDO WR DEDQGRQ ZKHQ

D �� LW PXVW DOVR EH RSWLPDO WR DEDQGRQ DW WKLV P ZKHQ D ! �� 7KH SURSRVLWLRQ DOVR SURYHV WKDW

LW LV VXERSWLPDO WR DEDQGRQ EHIRUH WKLV P�

:H DOVR GHULYH WKH RSWLPDO WLPH WR PRWKEDOO WKH SURMHFW� 7KLV UHTXLUHV D VHULHV RI OHPPDV� VWDWHG

DQG SURYHG LQ WKH DSSHQGL[� UHJDUGLQJ WKH VORSHV RI WKH DUJXPHQWV RI WKH WZR WHUPV LQ WKH UHFXUVLRQ

WKDW JLYHV WKH YDOXH IXQFWLRQ� HTXDWLRQ �� 7KHVH DUJXPHQWV VSHFLI\ WKH DQWLFLSDWHG YDOXH RI WKH

SURMHFW JLYHQ WKDW LQYHVWPHQW FRQWLQXHV� DQG LWV YDOXH ZKHQ LQYHVWPHQW LV VXVSHQGHG� 7KH OHPPDV

VKRZ WKDW WKH YDOXH FRQGLWLRQDO RQ FRQWLQXHG LQYHVWPHQW LV DOZD\V PRUH UHVSRQVLYH WR WKH YDOXH RI

WKH XQGHUO\LQJ FDVK oRZV� 7KXV� IRU n[HG Q DQG P WKHUH LV D XQLTXH YDOXH RI p�W� DW ZKLFK WKH

nUP LV LQGLmHUHQW WR LQYHVWLQJ RU QRW� 2SWLPDO SROLFLHV FDQ EH FKDUDFWHUL]HG LQ WHUPV RI WKLV YDOXH�

:H H[SORLW WKH IDFW WKDW WKH FULWLFDO YDOXH RI p�W� LV LQFUHDVLQJ LQ WKH n[HG FRVW LQ RXU QXPHULFDO

DQDO\VLV�

��

3URSRVLWLRQ �� 2SWLPDO ,QYHVWPHQW 3ROLF\� :KHQ FRVWV KDYH ERWK SURSRUWLRQDO DQG n[HG

FRPSRQHQWV �D ! �� E ! �� WKHQ IRU Pe�Q� GHnQHG LQ 3URSRVLWLRQ �� WKH SURMHFW LV DEDQGRQHG LI

DQG RQO\ LI P ! Pe�Q�� ,I P � Pe�Q�� WKHQ WKHUH LV D XQLTXH pe�Q�P� VXFK WKDW WKH SURMHFW LV

PRWKEDOOHG LI DQG RQO\ LI p�W� � pe�Q�P�� )XUWKHUPRUH� pe�Q�P� LV GHFUHDVLQJ LQ D� VR LI WKH n[HG

FRVWV DUH LQFUHDVHG SURMHFWV DUH PRWKEDOOHG DW ORZHU OHYHOV RI p�W��

7KHUH DUH WZR ZD\V D SURMHFW FDQ EH PRWKEDOOHG_HLWKHU D QHJDWLYH UHVROXWLRQ RI XQFHUWDLQW\

FDXVHV p WR GURS EHORZ pe� RU UHSHDWHG IDLOXUH FDXVHV pe WR ULVH DERYH WKH FXUUHQW p� +RZHYHU� WKHUHLV RQO\ RQH ZD\ D SURMHFW FDQ EH DEDQGRQHG_WKURXJK UHSHDWHG IDLOXUH� $ QHJDWLYH UHVROXWLRQ RI

p XQFHUWDLQW\ FDQQRW OHDG WR DEDQGRQPHQW EHFDXVH Pe LV QRW D IXQFWLRQ RI p� 2I FRXUVH� RQFH WKHSURMHFW LV DEDQGRQHG LWV YDOXH GURSV WR ]HUR ZKLFK LPSOLHV WKDW OLPPb!� 9 �Q�P� p� ��

�� 5LVN 3UHPLD

,Q WKLV VHFWLRQ ZH VWXG\ WKH UHTXLUHG H[SHFWHG UHWXUQ HDUQHG RQ DQ RZQHUVKLS FODLP WR WKH 5'

YHQWXUH� :H DUH LQWHUHVWHG ERWK LQ KRZ WKH V\VWHPDWLF DQG LGLRV\QFUDWLF ULVNV RI WKH YHQWXUH LQWHUDFW

WR GHWHUPLQH WKLV UHTXLUHG UHWXUQ� DQG LQ WKH GHSHQGHQFH RI WKH UHTXLUHG UHWXUQ RQ KRZ FORVH WKH

SURMHFW LV WR FRPSOHWLRQ� ,W VHHPV LQWXLWLYH WKDW D SURMHFW FORVH WR FRPSOHWLRQ VKRXOG LQKHULW WR D

JUHDWHU GHJUHH WKH ULVN FKDUDFWHULVWLFV RI WKH VWRFKDVWLF FDVK oRZ VWUHDP LW ZLOO XOWLPDWHO\ GHOLYHU�

7KRVH IDU IURP FRPSOHWLRQ PLJKW EH H[SHFWHG WR EHKDYH TXLWH GLmHUHQWO\�

7KLV DQDO\VLV PD\ EH KHOSIXO LQ DSSOLFDWLRQV ZKHUH FKRLFHV PXVW EH PDGH DERXW WKH GLVFRXQW

UDWH RU FRVW RI FDSLWDO DSSURSULDWH IRU HYDOXDWLQJ 5' H[SHQGLWXUHV� 6WDQGDUG SUDFWLFH DSSHDUV

WR EH WR DSSO\ D FRQVWDQW GLVFRXQW UDWH WR ?SUREDELOLW\ ZHLJKWHG� FDVK oRZV LQ FDOFXODWLQJ SUHVHQW

YDOXHV� 7KH GLVFRXQW UDWH HPSOR\HG LV JHQHUDOO\ WKH nUPV HVWLPDWHG FRVW RI FDSLWDO� RU D FRVW RI

FDSLWDO DSSURSULDWH WR WKH LQGXVWU\ LQ ZKLFK WKH LQYHVWPHQW LV EHLQJ PDGH�

0\HUV DQG +RZH �� DUJXH WKDW WKH FRVW RI FDSLWDO VKRXOG GHFUHDVH WKURXJK WKH OLIH RI WKH

SURMHFW� GXH WR WKH KLJKHU ?OHYHUDJH� RI WKH SURMHFW HDUO\ LQ LWV OLIH� $W nUVW� PXFK RI WKH SUHVHQW

YDOXH RI WKH SURMHFW LV DVVRFLDWHG ZLWK WKH UHODWLYHO\ n[HG 5' LQYHVWPHQWV� DV RSSRVHG WR WKH

YDULDEOH QHW UHYHQXHV WKDW EHJLQ ZKHQ WKH SURMHFW LV nQLVKHG� $QDORJRXV WR D OHYHUHG nUP� WKH

UHTXLUHG UHWXUQ RQ WKH ?HTXLW\� LQ WKH SURMHFW VKRXOG EH KLJKHU ZKHQ LW LV PRUH OHYHUHG� $V WKH

5' LV EURXJKW WR FRPSOHWLRQ� DQG WKH n[HG FODLPV DUH ?SDLG GRZQ�� WKH UHTXLUHG UHWXUQ VKRXOG

IDOO� 2XU PRGHO SURYLGHV D QDWXUDO ODERUDWRU\ LQ ZKLFK WR HYDOXDWH WKHVH LQWXLWLRQV DQG TXDQWLI\

WKHLU LPSRUWDQFH�

7KH PRGHO SURYLGHV D SDUWLFXODUO\ VLPSOH VHWWLQJ LQ ZKLFK WR DGGUHVV WKHVH TXHVWLRQV EHFDXVH

WKH ULVN SUHPLXP HDUQHG E\ WKH XQGHUO\LQJ FDVK oRZ SURFHVV LV FRQVWDQW� DV 3URSRVLWLRQ � VKRZV�

7KXV� DQ\ YDULDELOLW\ LQ WKH UHTXLUHG UHWXUQ PXVW EH GXH WR FKDUDFWHULVWLFV RI WKH 5' LWVHOI�

6LQFH ZH FDQQRW VROYH H[SOLFLWO\ IRU WKH YDOXH IXQFWLRQ DQG WKXV WKH ULVN SUHPLXP� RXU DELOLW\

WR DGGUHVV WKHVH TXHVWLRQV DQDO\WLFDOO\ LV OLPLWHG� :H FDQ OHDUQ D JRRG GHDO DERXW WKH VRXUFHV RI

YDULDWLRQ LQ WKH ULVN SUHPLXP� KRZHYHU� E\ FRQVLGHULQJ FDVHV ZKHUH LW LV FRQVWDQW� $Q LQFRPSOHWH

SURMHFW GLmHUV IURP D FRPSOHWHG RQH IRU IRXU UHDVRQV� )LUVW� WKH VWUHDP RI FDVK oRZV LV UDQGRPO\

��

GHOD\HG� 6HFRQG� WKH nUP LQFXUV n[HG FRVWV SULRU WR FODLPLQJ WKH UDQGRP FDVK oRZ VWUHDP� 7KXV�

WKH nUP LV ?VKRUW� D n[HG FODLP DQG ?ORQJ� D YDULDEOH FODLP� 7KLV FRQIHUV RSHUDWLQJ OHYHUDJH RQ WKH

SURMHFW� 7KLUG� WKH nUP KROGV DQ RSWLRQ WR GHOD\ WKHVH H[SHQGLWXUHV� )RXUWK� LW DOVR KDV DQ RSWLRQ

WR DEDQGRQ WKH SURMHFW DOWRJHWKHU� $V 3URSRVLWLRQ �� PDNHV FOHDU� WKH GHFLVLRQ WR DEDQGRQ YHUVXV

WKH GHFLVLRQ WR PRWKEDOO FDQ EH YLHZHG DV TXDOLWDWLYHO\ GLmHUHQW LQ WKDW WKH GHFLVLRQ WR DEDQGRQ LV

PDGH SXUHO\ LQ UHVSRQVH WR WHFKQRORJLFDO �LGLRV\QFUDWLF� ULVN� 7KH ?FXWRm UXOH� DW ZKLFK WKH SURMHFW

KDV ]HUR FRQWLQXDWLRQ YDOXH LV RI WKH IRUP Pe�Q�� DQG GRHV QRW GHSHQG RQ p�W�� 7KH GHFLVLRQ WR

VXVSHQG LQYHVWPHQW� RQ WKH RWKHU KDQG� GHSHQGV RQ LQIRUPDWLRQ DERXW IXWXUH FDVK oRZV� DQG WKHVH

KDYH V\VWHPDWLF ULVN�

:KHQ FRVWV DUH SURSRUWLRQDO� WKHQ WKH RSWLRQ WR PRWKEDOO �UDWKHU WKDQ DEDQGRQ� KDV QR YDOXH�

DQG WKH OHYHUDJH DVVRFLDWHG ZLWK WKH n[HG FRPSRQHQW RI WKH FRVWV LV DEVHQW� 7KH QH[W SURSRVLWLRQ

VKRZV WKDW LQ WKLV FDVH WKH ULVN SUHPLXP RQ WKH SURMHFW LV WKH VDPH DV WKDW HDUQHG RQ WKH FDVK

oRZV� DQG WKXV WKDW WKH VRXUFHV RI YDULDWLRQ LQ WKH ULVN SUHPLXP DUH DWWULEXWDEOH WR SUHFLVHO\ WKHVH

IDFWRUV�

3URSRVLWLRQ �� :KHQ WKHUH DUH QR n[HG 5' FRVWV �D �� WKH ULVN SUHPLXP RI WKH YHQWXUH LV

LQGHSHQGHQW RI Q RU P� DQG HTXDO WR w� WKH ULVN SUHPLXP RI WKH FRPSOHWHG SURMHFW�

$ IXUWKHU VSHFLDOL]DWLRQ RI WKH PRGHOV DOORZV XV WR DVN ZKDW HmHFW SXUH UDQGRP GHOD\ KDV RQ

ULVN SUHPLD LQ D FDSLWDO EXGJHWLQJ VHWWLQJ� $ VSHFLDO FDVH RI 3URSRVLWLRQ �� LV ZKHQ DOO 5' FRVWV

DUH ]HUR� VR WKDW WKH ?5'� LV MXVW D GHOD\ LQ WKH DUULYDO RI D VWUHDP RI FDVK oRZV� %HFDXVH WKH

IDFWRUV FRQWULEXWLQJ WR GHOD\ DUH SXUHO\ LGLRV\QFUDWLF� RQH PLJKW WKLQN WKDW WKH V\VWHPDWLF ULVN RI

WKH 5' ZRXOG EH OHVV WKDQ WKDW RI WKH FDVK oRZV WKHPVHOYHV� 3URSRVLWLRQ �� VKRZV WKDW WKLV

LV QRW WKH FDVH� +LJKHU SUREDELOLW\ RI GHOD\ UHGXFHV ERWK WKH YDOXH WRGD\ DQG WKH H[SHFWDWLRQ RI

IXWXUH YDOXH SURSRUWLRQDOO\� DV ORQJ DV WKH GHOD\ LV LGLRV\QFUDWLF� VR WKDW WKH H[SHFWHG UHWXUQ LV

XQDmHFWHG��

7KH IDFW WKDW WKH GHFLVLRQ WR DEDQGRQ GRHV QRW DmHFW WKH ULVN SUHPLXP LV QRW RQO\ D SURSHUW\

RI WKLV VSHFLDO FDVH� DQG LW LOOXVWUDWHV DQ LPSRUWDQW SRLQW� 7KH IDFWRUV WKDW LPSDUW V\VWHPDWLF ULVN

WR D YHQWXUH DUH QDWXUDOO\ YLHZHG DV H[RJHQRXV WR WKH 5' SURFHVV LWVHOI� ,QIRUPDWLRQ DERXW WKHVH

H[RJHQRXV IDFWRUV FRQWLQXHV WR HYROYH LQGHSHQGHQW RI WKH nUPV LQYHVWPHQW FKRLFHV� )RU H[DPSOH�

PDFURHFRQRPLF FRQGLWLRQV FDQ EH REVHUYHG ZLWKRXW GLUHFW LQYHVWPHQW E\ WKH nUP� 7KXV� WKH

UHVSRQVH WR EDG QHZV DERXW VXFK IDFWRUV LV WR GHOD\ LQYHVWPHQW DQG VHH LI FRQGLWLRQV LPSURYH� 7KH

GHFLVLRQ WR ZDON DZD\ IURP D SURMHFW LV PXFK PRUH OLNHO\ WR EH GULYHQ E\ D GHWHUPLQDWLRQ WKDW LW

LV WHFKQRORJLFDOO\ LQIHDVLEOH RU SURKLELWLYHO\ FRVWO\� DQG WKH IDFWRUV OHDGLQJ WR VXFK D GHWHUPLQDWLRQ

DUH OLNHO\ WR EH LGLRV\QFUDWLF WR WKH SURMHFW� nUP� RU LQGXVWU\�

�%\ ZD\ RI DQDORJ\� FRQVLGHU D ORWWHU\ WKDW� ZLWK D n[HG SUREDELOLW\� VLPSO\ GHOLYHUV WKH 63 �� WR WKH ZLQQHUDW GDWH W� ~ � 7KH DUJXPHQWV LQ WKH SURRI RI 3URSRVLWLRQ �� WHOO XV WKDW WKLV ORWWHU\ ZLOO EH SULFHG DW GDWH W WR \LHOGWKH VDPH H[SHFWHG UHWXUQ DV WKH 63� $ PRUH FRPSOLFDWHG ORWWHU\ WKDW RmHUHG GLmHUHQW UDQGRPO\ VL]HG SRVLWLRQV LQWKH 63� DW D UDQGRP IXWXUH GDWH� ZRXOG DOVR EH SULFHG WKLV ZD\� 6LQFH WKH YDULDEOHV WKDW GHWHUPLQH H[HUFLVH RI WKHRSWLRQV HPEHGGHG LQ 5' DUH H[HUFLVHG LQGHSHQGHQWO\ RI Ip�W�� W � 7 J ZKHQ FRVWV DUH SURSRUWLRQDO� WKH SURMHFWVSD\RmV DUH VLPLODU WR WKRVH RI VXFK D ORWWHU\�

��

2XU PRGHO LOOXVWUDWHV WKHVH GLVWLQFWLRQV LQ D SDUWLFXODUO\ VWDUN PDQQHU� %\ 3URSRVLWLRQ �� WKH

GHFLVLRQ WR DEDQGRQ WKH YHQWXUH GRHV QRW GHSHQG RQ p� WKDW LV� LW LV D IXQFWLRQ RI RQO\ WHFKQRORJLFDO

ULVN ZKLFK LV LGLRV\QFUDWLF� 7KXV� WKLV RSWLRQ ZLOO QHYHU FRPPDQG D ULVN SUHPLXP� %\ WKH VDPH

WRNHQ� WKH RSWLPDO H[HUFLVH RI WKH RSWLRQ WR PRWKEDOO GRHV GHSHQG RQ p �E\ WKH VDPH SURSRVLWLRQ��

DQG VR 5' SURMHFWV LQ ZKLFK WKLV RSWLRQ LV YDOXDEOH ZLOO FRPPDQG ULVN SUHPLXP LQ H[FHVV RI w�

2I FRXUVH� HYHQ WKRXJK WKH nUPV LQYHVWPHQW H[SHULHQFH LV LGLRV\QFUDWLF� LW VWLOO DmHFWV WKH

QDWXUH RI WKH nUPV FXUUHQW SRVLWLRQ LQ WKH V\VWHPDWLF YDULDEOH� DQG LWV H[SRVXUH WR WKLV ULVN� 7KXV�

LQ WKH JHQHUDO FDVH� ZH nQG WKDW RSWLPDO H[HUFLVH RI WKH RSWLRQ WR PRWKEDOO GHSHQGV RQ Q DQG P�

$ FKDQJH LQ HLWKHU Q RU P ZLOO DmHFW WKH ?LQ WKH PRQH\QHVV� RI WKH RSWLRQ� DQG WKH ULVN SUHPLXP

RI DQ RSWLRQ LV DmHFWHG E\ KRZ PXFK LW LV LQ RU RXW RI WKH PRQH\� 7KLV LPSOLHV LV WKDW WKH DPRXQW

E\ ZKLFK WKH ULVN SUHPLXP RI WKH LQFRPSOHWH SURMHFW H[FHHGV w LV DmHFWHG E\ WKH UHVROXWLRQ RI

WHFKQLFDO XQFHUWDLQW\� HYHQ WKRXJK WKLV ULVN LV SXUHO\ LGLRV\QFUDWLF DQG VR LV QRW SULFHG� ,QWXLWLYHO\�

WKH UHDVRQ IRU WKLV LV WKDW RQH FDQ WKLQN RI WKH WRWDO H[SHFWHG FRVWV RI WKH 5' DV WKH ?VWULNH�

SULFH RI WKH JURZWK RSWLRQ� &RPSOHWLQJ D VWDJH UHGXFHV WKLV H[SHFWHG FRVW� ZKLFK ORZHUV WKH VWULNH

SULFH RI WKH RSWLRQ DQG WKH ULVN SUHPLXP�

$ QDWXUDO ZD\ WR WKLQN RI WKH 5' YHQWXUH� WKHQ� LV DV D VHULHV RI FRPSRXQG RSWLRQV RQ WKH

XQGHUO\LQJ FDVK oRZV� 7KLV DQDORJ\ VXJJHVWV WKDW ZKHQ WKH RSWLRQ YDOXH LV D IXQFWLRQ RI V\VWHPDWLF

ULVN� WKH SURMHFW ZLOO KDYH KLJKHU UHTXLUHG UHWXUQV DV ZH PRYH EDFN WKURXJK LWV VWDJHV� :H QH[W

VKRZ� QXPHULFDOO\� WKDW WKLV LV LQGHHG WKH FDVH�

� 1XPHULFDO 5HVXOWV

,Q WKLV VHFWLRQ ZH XVH QXPHULFDO PHWKRGV WR DQDO\]H LQ JUHDWHU GHWDLO VRPH RI WKH SURSHUWLHV RI

WKH 5' YHQWXUH� :H FRQVLGHU WZR SDUDPHWHUL]DWLRQV� )LUVW� ZH FDOFXODWH YDOXHV DQG ULVN SUHPLD

DVVXPLQJ WKDW { LV NQRZQ� 7KLV VSHFLDO FDVH SURYLGHV D VLPSOH HQYLURQPHQW LQ ZKLFK WKH LQFUHPHQWDO

LPSRUWDQFH RI WKH RSWLRQ WR PRWKEDOO FDQ EH JDXJHG� 7KH VHFRQG SDUDPHWHUL]DWLRQ LV WKH JHQHUDO

PRGHO DQG DOORZV DQ DVVHVVPHQW RI KRZ WKH ULVN SUHPLXP LV DmHFWHG E\ WKH UHVROXWLRQ RI XQFHUWDLQW\

DERXW WKH nUPV UHVHDUFK SURGXFWLYLW\� 6LQFH LQ WKH JHQHUDO PRGHO LW PLJKW EH RSWLPDO WR DEDQGRQ�

WKLV FDVH DOVR DOORZV XV WR LQYHVWLJDWH ZK\ WKH RSWLRQ WR PRWKEDOO FRPPDQGV D ULVN SUHPLXP� ZKLOH

WKH RSWLRQ WR DEDQGRQ GRHV QRW�

:H SDUDPHWHUL]H WKH PRGHO WR TXDUWHUO\ GDWD� D SHULRG FRUUHVSRQGV WR WKUHH PRQWKV� 7DEOH �

OLVWV WKH YDOXHV RI WKH SDUDPHWHUV HPSOR\HG� 7KH SURMHFW ZRXOG WDNH nYH \HDUV WR FRPSOHWH LI

HYHU\ LQYHVWPHQW ZHUH VXFFHVVIXO� 7KH GLYLGHQG SURFHVV JURZV DW �� SHU \HDU RQ DYHUDJH� ZLWK DQ

DQQXDO VWDQGDUG GHYLDWLRQ RI DERXW �� VR LWV LQQRYDWLRQV DUH TXLWH YDULDEOH UHODWLYH WR WKLV JURZWK

UDWH� 7KH nUPV DQG WKH PDUNHWV SULRUV DERXW WKH FKDQFH RI VXFFHVV DW HDFK VWDJH DUH GLmXVH DQG

XQLIRUP� D VSHFLDO FDVH RI WKH EHWD ZLWK ERWK SDUDPHWHUV HTXDO WR RQH� $OWHUQDWLYHO\� ZKHQ { LV

NQRZQ� VXFFHVV RFFXUV DW HDFK VWDJH ZLWK D SUREDELOLW\ RI �� ZKLFK LV DOVR WKH LQLWLDO SULRU LQ WKH

XQNQRZQ FDVH� 7KH SUREDELOLW\ RI VXUYLYLQJ RQH \HDU ZLWKRXW REVROHVFHQFH �{�\� LV �� 7KH n[HG

FRVWV DUH DVVXPHG WR EH �� SHU SHULRG ZKLOH WKH YDULDEOH FRVWV DUH �� 7KXV IRU YDOXHV RI p DURXQG

��

3DUDPHWHU { XQNQRZQ { NQRZQ

1 �� x �� }p �� U �� }] �� { ^ ��m� �� ^m� �� ^{\ �� D �� E �� w �� w ��

7DEOH �� 3DUDPHWHU 9DOXHV IRU 6LPXODWLRQV

WHQ WKH n[HG DQG YDULDEOH FRVWV FRQWULEXWH HTXDOO\ WR WKH WRWDO FRVWV� 7KH YDOXH RI WKH VWDQGDUG

GHYLDWLRQ LQ WKH SULFLQJ NHUQHO �DERXW �� SHU \HDU� LV FKRVHQ� WRJHWKHU ZLWK WKH FRUUHODWLRQ EHWZHHQ

WKH SULFLQJ NHUQHOV LQQRYDWLRQV DQG WKRVH RI WKH FDVK oRZ SURFHVV �� WR SURYLGH D w �� 2Q DQ

DQQXDO EDVLV WKLV FRUUHVSRQGV WR D ULVN SUHPLXP RI �� IRU WKH FDVK oRZV RI WKH FRPSOHWHG SURMHFW�

6LQFH WKH ULVN IUHH UDWH LV �� SHU \HDU� WKLV LPSOLHV DQ H[SHFWHG UHWXUQ RI �� IRU WKH FRPSOHWHG

SURMHFW� 7KH nQDO OLQH LQ 7DEOH � JLYHV WKH XSSHU ERXQG ��w� IRU WKH ULVN SUHPLXP� GHULYHG EHORZ

LQ 3URSRVLWLRQ �� 7KLV LV �� RU �� RQ DQ DQQXDO EDVLV� ZKLFK LPSOLHV DQ XSSHU ERXQG RQ WKH

H[SHFWHG UHWXUQ RI WKH LQFRPSOHWH SURMHFW RI �� DQQXDOO\�

�� .QRZQ 3UREDELOLW\ RI 6XFFHVV

7KH FDVH ZKHUH { LV NQRZQ WR WKH nUP SURYLGHV D XVHIXO EHQFKPDUN IRU VHYHUDO UHDVRQV� ,Q WKLV

FDVH DOO XQFHUWDLQW\ LV DWWULEXWDEOH WR MXVW WZR VRXUFHV� WKH FKDUDFWHULVWLFV RI WKH FDVK oRZ DQG WKH

SXUH XQFHUWDLQW\ RI RYHUFRPLQJ WHFKQLFDO KXUGOHV� 7KLV DOORZV XV WR DQDO\]H WKH HmHFWV WKDW DUH QRW

D IXQFWLRQ RI WKH XQFHUWDLQW\ DERXW WHFKQLFDO DELOLW\�

:LWK SURSRUWLRQDO 5' FRVWV� WKH RSWLRQ WR PRWKEDOO LV QHYHU H[HUFLVHG DQG VR WKH LQFRPSOHWH

SURMHFW KDV WKH VDPH ULVN SUHPLXP DV WKH FRPSOHWHG SURMHFW� :KHQ WKHUH LV D n[HG FRPSRQHQW LQ

WKH 5' FRVW� WKH RSWLRQ WR PRWKEDOO KDV YDOXH DQG FRPPDQGV D ULVN SUHPLXP� 7KH YROXQWDU\

QDWXUH RI WKH LQYHVWPHQW GHFLVLRQ� ZKLFK LV PDGH FRQGLWLRQLQJ RQ FXUUHQW YDOXHV RI p�W� LPSDUWV

V\VWHPDWLF ULVN WR WKH SURMHFW� ,Q DGGLWLRQ� WKH nUPV n[HG LQYHVWPHQW H[SHQGLWXUHV FDQ EH YLHZHG

DV D OLDELOLW\ WKDW RmVHWV WKH YDOXH RI LWV DVVHW� WKH FDVK oRZ VWUHDP� 6LQFH WKH n[HG OLDELOLW\ KDV

ORZHU ULVN WKDQ WKH VWRFKDVWLF DVVHW� WKH nUP KDV RSHUDWLQJ OHYHUDJH WKDW ZLOO DOVR DOWHU WKH ULVN

SUHPLXP� 7R JDXJH WKH UHODWLYH LPSRUWDQFH RI WKHVH WZR HmHFW� ZH FRPSXWH WKH YDOXH DQG H[SHFWHG

UHWXUQ RI WKH LQFRPSOHWH SURMHFW ZLWK DQG ZLWKRXW WKH RSWLRQ WR PRWKEDOO� 7KDW LV� ZH FRPSXWH

��

WKH YDOXH RI WKH SURMHFW DV LQ RXU PRGHO� :H DOVR FRPSXWH LW DVVXPLQJ WKH nUP PXVW LQYHVW HDFK

SHULRG� &ORVHG IRUP H[SUHVVLRQV IRU WKH YDOXH LQ WKLV FDVH DUH SURYLGHG LQ /HPPD �� ZKLFK LV VWDWHG

DQG SURYHG LQ WKH $SSHQGL[�

7DEOH � VKRZV WKH YDOXH RI WKH 5' SURMHFW IRU GLmHUHQW OHYHOV RI p DW GLmHUHQW VWDJHV RI

GHYHORSPHQW� 7KH YDOXHV LQ SDUHQWKHVLV LV WKH YDOXH RI WKH YHQWXUH DVVXPLQJ LQYHVWPHQW LV QRQ�

YROXQWDU\� 7KXV� WKH GLmHUHQFH EHWZHHQ WKH DFWXDO YDOXHV DQG WKH YDOXHV LQ SDUHQWKHVHV LV D PHDVXUH

RI KRZ LPSRUWDQW WKH RSWLRQ� DV RSSRVHG WR MXVW WKH RSHUDWLQJ OHYHUDJH DVVRFLDWHG ZLWK WKH n[HG

FRVWV� LV� 7KH YDOXH RI WKLV RSWLRQ LV PRVW LPSRUWDQW LQ WKH HDUO\ VWDJHV RI GHYHORSPHQW DQG ORZHU

YDOXHV RI p� 7KH LPSRUWDQFH RI WKH FRQYH[LW\ WKDW WKLV LPSDUWV FDQ EH JDXJHG E\ FRPSDULQJ WKH

SULFH DSSUHFLDWLRQ IRU DQ LQFUHDVH LQ p ZKHQ WKH RSWLRQ LV LPSRUWDQW �H�J�� FRPSOHWHG VWDJHV� WR

ZKHQ LW LV QRW �H�J� �� FRPSOHWHG VWDJHV��

1XPEHU RI &RPSOHWHG 6WDJHV �Q�p � � � ��

9DOXH ��

�5LVN �� 3UHP �^� �^� ��

9DOXH ��

��5LVN �� 3UHP �^� �^� ��

9DOXH ��

��5LVN �� 3UHP �^� ��

9DOXH ��

��5LVN �� 3UHP �^� ��

7DEOH �� 9DOXH DQG 5LVN 3UHPLXP RI WKH 5' 9HQWXUH ZKHQ { LV .QRZQ� 7KH FHOOV LQ WKHWDEOH VKRZ WKH YDOXH DQG H[SHFWHG H[FHVV UHWXUQ RI D YHQWXUH WKDW UHTXLUHV �� VWDJHV WR FRPSOHWH IRU D JLYHQ

p DQG QXPEHU RI FRPSOHWHG VWDJHV� %ROG IDFH YDOXHV FRUUHVSRQG WR PRWKEDOOHG SURMHFWV� 7KH QXPEHUV LQ

SDUHQWKHVHV JLYH WKH YDOXH DQG H[SHFWHG UHWXUQ RI WKH YHQWXUH DVVXPLQJ GHYHORSPHQW PXVW EH XQGHUWDNHQ�

WKDW LV� WKH RSWLRQ WR PRWKEDOO GRHV QRW H[LVW�

6LQFH p KDV D V\VWHPDWLF FRPSRQHQW� WKH QRQOLQHDULWLHV FORVH WR WKH SRLQW ZKHUH WKH SURMHFW LV

PRWKEDOOHG DUH DOVR UHoHFWHG LQ WKH ULVN SUHPLXP� ,Q 7DEOH � WKH ERWWRP WZR QXPEHUV LQ HDFK FHOO

DUH WKH ULVN SUHPLXP RQ WKH SURMHFW DQG� LQ SDUHQWKHVHV� ZKDW WKH ULVN SUHPLXP ZRXOG EH LI WKHUH

ZHUH QR RSWLRQ WR DEDQGRQ RU PRWKEDOO� &RPSDULQJ WKHVH QXPEHUV DOORZV XV WR VHSDUDWHO\ HYDOXDWH

WKH FRQWULEXWLRQ RI RSHUDWLQJ OHYHUDJH DQG WKH YROXQWDU\ H[HUFLVH GHFLVLRQ WR WKH ULVN SUHPLXP�

��

)RU KLJK YDOXHV RI p DQG ZKHQ WKH SURMHFW LV FORVH WR FRPSOHWLRQ� WKHVH YDOXHV DUH FORVH WRJHWKHU

HYHQ WKRXJK WKH ULVN SUHPLXP H[FHHGV WKDW HDUQHG RQ WKH FRPSOHWHG SURMHFW� 7KH ULVN SUHPLXP LQ

WKHVH FDVHV LV DWWULEXWDEOH WR WKH RSHUDWLQJ OHYHUDJH DVVRFLDWHG ZLWK WKH n[HG 5' H[SHQVHV� UDWKHU

WKDQ WKH YROXQWDU\ QDWXUH RI WKRVH H[SHQVHV� ,Q WKH UHJLRQ ZKHUH WKH nUP LV FORVH WR H[HUFLVLQJ WKH

RSWLRQ WR PRWKEDOO� RQ WKH RWKHU KDQG� WKH ULVN SUHPLXP RQ WKH SURMHFW LV ORZHU WKDQ WKDW RQ WKH

DQDORJRXV YHQWXUH ZLWK GHWHUPLQLVWLF H[SHQGLWXUH� 7KH RSWLRQ WR VXVSHQG LQYHVWPHQW WUXQFDWHV WKH

GRZQVLGH RI WKH GLVWULEXWLRQ� ORZHULQJ ERWK V\VWHPDWLF DQG XQV\VWHPDWLF ULVN� 7KXV� WKH ?OHYHUDJH�

DQG WKH ?RSWLRQDOLW\� HPEHGGHG LQ WKH 5' YHQWXUH RSHUDWH LQ RSSRVLWH GLUHFWLRQV� 7KLV LV QRW

VXUSULVLQJ ZKHQ RQH FRQVLGHUV DQ DQDORJ\ IURP WUDGLWLRQDO RSWLRQ SULFLQJ WKHRU\� $Q RSWLRQ ZKLFK

LV LQ WKH PRQH\ HDUQV D ORZHU ULVN SUHPLXP WKDQ D IRUZDUG FRQWUDFW RQ WKH VDPH XQGHUO\LQJ DVVHW

ZKHQ WKH VWULNH SULFH DQG IRUZDUG SULFH DUH HTXDO��

)LJXUH � SORWV WKH ULVN SUHPLXP �� SHU \HDU� DJDLQVW WKH QXPEHU RI VWDJHV FRPSOHWHG� Q� DQG

WKH YDOXH RI WKH FDVK oRZ SURFHVV� p� 7KH GDUN OLQH LV pe�Q� �IRU D JLYHQ Q� WKH KLJKHVW p IRU

ZKLFK WKH SURMHFW LV VWLOO PRWKEDOOHG�� 7KH ULVN SUHPLXP LV KLJKHVW ZKHQ WKH SURMHFW LV PRWKEDOOHG�

7KH DUHD EHORZ WKH oDW UHJLRQ LV WKH UHJLRQ ZKHQ DFWLYH GHYHORSPHQW RFFXUV� ,W LV FOHDU IURP

WKH SORW WKDW WKH ULVN SUHPLXP GURSV GUDPDWLFDOO\ DV VRRQ DV HFRQRPLF FRQGLWLRQV LPSURYH WR WKH

SRLQW WKDW WKH 5' LV XQGHUWDNHQ� ,W FRQWLQXHV WR IDOO WRZDUGV WKH ULVN SUHPLXP HDUQHG E\ WKH

FRPSOHWHG SURMHFW DV FLUFXPVWDQFHV LPSURYH DQG SURJUHVV FRQWLQXHV� 7KH ULVN SUHPLXP GHFUHDVHV

PRQRWRQLFDOO\ RYHU WKH SURMHFWV OLIH� FRQVLVWHQW ZLWK WKH DUJXPHQWV RI 0\HUV DQG +RZH ��

WKDW WKH LPSOLFLW OHYHUDJH LQ WKH SURMHFW IDOOV DV LW SURFHHGV� DQG ZLWK WKH LQWXLWLRQ WKDW WKH SURMHFW

LV D FRPSRXQG RSWLRQ RQ WKH XQGHUO\LQJ FDVK oRZ SURFHVV� $Q LPSRUWDQW LPSOLFDWLRQ RI WKLV LV

WKDW WKH UHVROXWLRQ RI WHFKQLFDO XQFHUWDLQW\ FDQ GUDPDWLFDOO\ DmHFW WKH ULVN SUHPLXP� 7KLV LV WUXH

GHVSLWH WKH IDFW WKDW WKH WHFKQLFDO XQFHUWDLQW\ LV SXUHO\ LGLRV\QFUDWLF� )LQDOO\� LW LV HYLGHQW WKDW

WKLV PRQRWRQLF GHFUHDVH LQ WKH ULVN SUHPLXP LV QRW D VWHDG\� OLQHDU RQH�

7KH ORZHU ERXQG WKH ULVN SUHPLXP DSSURDFKHV DV WKH SURMHFW PDWXUHV LV WKH ULVN SUHPLXP HDUQHG

E\ WKH FDVK oRZ SURFHVV LWVHOI� $V p�W� ULVHV� WKH ULVN SUHPLXP DSSURDFKHV WKLV OHYHO DW HDUOLHU SRLQWV

LQ WKH SURMHFWV OLIH� 7KLV LV QRW VXUSULVLQJ� VLQFH DV ZH VDZ LQ 6HFWLRQ �� WKH YDOXH RI WKH 5'

SURMHFW DSSURDFKHV SURSRUWLRQDOLW\ LQ p�W�� DV p�W� ULVHV DQG WKH DEDQGRQPHQW RSWLRQ EHFRPHV OHVV

LPSRUWDQW� )LQDOO\� QRWH WKH VHQVLWLYLW\ RI pe�Q�� WKH SRLQW MXVW EHIRUH DFWLYH GHYHORSPHQW EHJLQV�WR WKH QXPEHU RI FRPSOHWHG VWDJHV HDUO\ RQ LQ GHYHORSPHQW� %\ FRPSOHWLQJ MXVW WKH nUVW VWDJH LW

GURSV IURP �� WR �� ,Q FRQWUDVW� FRPSOHWLQJ WKH IRXUWHHQWK VWDJH KDUGO\ FKDQJHV pe�Q��,W LV FOHDU IURP LQVSHFWLRQ RI )LJXUH � WKDW WKH ULVN SUHPLXP DSSURDFKHV� DV D ORZHU ERXQG� WKH

SUHPLXP HDUQHG RQ WKH FRPSOHWHG SURMHFW� ,W DOVR DSSHDUV WR oDWWHQ RXW DQG DSSURDFK DQ XSSHU

ERXQG LQ WKH PRWKEDOOHG UHJLRQ DV p DQG Q EHFRPH VPDOO� $ FKDUDFWHUL]DWLRQ RI WKLV XSSHU ERXQG

�&RQVLGHU D VHFXULW\ ZLWK D SD\Rm 6�7 � b . DW GDWH 7 ZKHUH 6�7 � LV WKH SULFH DW 7 RI D VWRFN WKDW SD\V QRGLYLGHQGV EHWZHHQ � DQG 7 � 7KH YDOXH DW W � 7 RI WKLV VHFXULW\ LV 9 �6�W�� W� 6�W�b. H[S�bU�7 b W�� %\ SXW�FDOOSDULW\ 9 �W� &�W�b3 �W� ZKHUH &�W� LV WKH SULFH RI D FDOO RQ WKH VWRFN DQG 3 �W� LV WKH SULFH RI D SXW RQ WKH VWRFN� ERWKZLWK H[HUFLVH SULFH . DQG ERWK (XURSHDQ RSWLRQV� 7KH ULVN SUHPLXP RQ WKH IRUZDUG FRQWUDFW LV HTXDO WR �mb U�966

9

DQG WKH ULVN SUHPLXP RI WKH FDOO LV �mb U�&66&� 6LQFH 96 �� 9 �W� &�W�b 3 �W� � &�W� DQG VLQFH &6 � �� LQ WKH

UHJLRQ ZKHUH 9 �W� ! �� WKH ULVN SUHPLXP RQ WKH IRUZDUG LV DOZD\V KLJKHU WKDQ WKH ULVN SUHPLXP RQ WKH FDOO�

��

0

4

8

12

16

n

10.

20.

30.

40.

50.

60.

70.

Delta

10

12.5

15

17.5

20

0

4

8

12

16

n

10

.5

5

)LJXUH �� 5LVN 3UHPLXP RI WKH 5' 9HQWXUH ZKHQ { LV NQRZQ� 7KH ULVN SUHPLXP �H[SUHVVHGLQ ��DQQXP� RI SURMHFW WKDW UHTXLUHV �� VWDJHV WR FRPSOHWH LV SORWWHG DV D IXQFWLRQ RI p� �ODEHOHG ?'HOWD�

LQ WKH SORW� DQG Q� WKH QXPEHU RI FRPSOHWHG VWDJHV� 7KH GDUN OLQH PDUNV� IRU D JLYHQ Q� WKH KLJKHVW p IRU

ZKLFK WKH SURMHFW LV VWLOO PRWKEDOOHG� WKDW LV� LW LV pe�Q�� 7KH ULVN SUHPLXP LV GHnQHG WR EH WKH GLmHUHQFH

EHWZHHQ WKH FRQWLQXRXVO\ FRPSRXQGHG H[SHFWHG UHWXUQ RI WKH SURMHFW DQG WKH ULVNOHVV UDWH�

��

PD\ EH XVHIXO LQ DSSOLFDWLRQV� VLQFH LW SURYLGHV D YHU\ FRQVHUYDWLYH ?KXUGOH UDWH� IRU WKH SURMHFW�

:KLOH ZH FDQQRW H[SOLFLWO\ VROYH IRU WKH YDOXH IXQFWLRQ RU WKH ULVN SUHPLXP LQ RXU GLVFUHWH�

WLPH PRGHO� WKH FRQWLQXRXV�WLPH DQDORJXH FDQ EH XVHG WR FKDUDFWHUL]H WKH SRLQW WKH ULVN SUHPLXP

DSSURDFKHV LQ WKH PRWKEDOOHG UHJLRQ� 6XSSRVH WKH YDOXHV RI p�W�� Q DQG P DUH FXUUHQWO\ VXFK WKDW LW

LV RSWLPDO QRW WR LQYHVW IXUWKHU LQ WKH SURMHFW� ,Q GLVFUHWH WLPH WKH SRVVLEOH YDOXHV RI p�W�� UHDFK

DFURVV UHJLRQV RI WKH VWDWH VSDFH ZKHUH LQYHVWPHQW RFFXUV DQG ZKHUH PRWKEDOOLQJ FRQWLQXHV� 6LQFH

WKH FDVK oRZ SURFHVV FDQQRW MXPS LQ FRQWLQXRXV WLPH� LW SRVVLEOH WR VROYH IRU WKH YDOXH IXQFWLRQ

?ORFDOO\� LQ WKH UHJLRQ ZKHUH PRWKEDOOLQJ RFFXUV� 2YHU WKLV UHJLRQ WKH ULVN SUHPLXP LV FRQVWDQW�

3URSRVLWLRQ �� ,Q WKH FRQWLQXRXV�WLPH FDVH� WKH YDOXH IXQFWLRQ LQ WKH UHJLRQ ZKHUH WKH SURMHFW LV

PRWKEDOOHG KDV WKH IRUP $�Q�p�� ZKHUH � LV D FRQVWDQW� DQG WKH ULVN SUHPLXP RQ WKH 5' SURMHFW

LV �w�

)LJXUH � VKRZV WKDW WKH ULVN SUHPLXP LV UHODWLYHO\ FRQVWDQW DQG DW LWV KLJKHVW LV LQ WKH UHJLRQ

ZKHUH LW LV PRWKEDOOHG� 7KH DERYH UHVXOW VKRZV WKDW WKLV EHKDYLRU LV DWWULEXWDEOH WR WKH FRQVWDQW

HODVWLFLW\ RI WKH YDOXH ZLWK UHVSHFW WR WKH VWDWH YDULDEOH WKDW KDV V\VWHPDWLF ULVN� p�W�� :H NQRZ

WKDW WKH FRQGLWLRQDO FRYDULDQFH EHWZHHQ WKH ORJ RI p�W� DQG WKH ORJ RI WKH SULFLQJ NHUQHO LV FRQVWDQW�

5DLVLQJ p�W� WR D SRZHU JUHDWHU WKDQ RQH NHHSV WKLV FRYDULDQFH FRQVWDQW� EXW LQFUHDVHV LW� �7KH

H[SUHVVLRQ IRU � GHULYHG LQ WKH DSSHQGL[ FDQ EH VKRZQ WR EH JUHDWHU WKDQ RQH�� 7KXV� WKH DQDO\VLV

SURYLGHV WKH ERXQGV ZH VHHN� 7KH YDOXH RI WKH 5' YHQWXUH LV PRVW HODVWLF ZLWK UHVSHFW WR

LQIRUPDWLRQ DERXW IXWXUH FDVK oRZV� DQG WKXV KDV WKH PRVW V\VWHPDWLF ULVN DQG WKH KLJKHVW ULVN

SUHPLXP� ZKHQ LW LV PRWKEDOOHG� ,W LV OHDVW HODVWLF ZLWK UHJDUG WR WKLV LQIRUPDWLRQ ZKHQ WKH SURMHFW

LV FRPSOHWH�

$Q LPSRUWDQW IHDWXUH RI WKH XSSHU ERXQG �w LV WKDW LW LV QRW D IXQFWLRQ RI WKH SUREDELOLW\ RI

VXFFHVV {� 7KLV LPSOLHV WKDW WKH DQDO\VLV FDQ DOVR EH DSSOLHG WR WKH FDVH ZKHQ { LV XQNQRZQ�

,W VHHPV TXLWH REYLRXV WKDW DV p�W� LQFUHDVHV DQG WKH SUREDELOLW\ WKDW WKH nUP ZLOO H[HUFLVH DOO

LWV RSWLRQV WR LQYHVW DSSURDFKHV XQLW\� WKH ULVN SUHPLXP VKRXOG DSSURDFK WKDW HDUQHG RQ WKH FDVK

oRZV� ,W LV OHVV REYLRXV WKDW WKH ULVN SUHPLXP LV ERXQGHG DERYH� 7KH DQDORJ\ WR WUDGLWLRQDO RSWLRQ

SULFLQJ WKHRU\ PD\ EH KHOSIXO KHUH� ,Q WKH %ODFN�6FKROHV PRGHO� WKH ULVN SUHPLXP RQ WKH FDOO RSWLRQ

LV HTXDO WR WKH HODVWLFLW\ RI WKH FDOO SULFH ZLWK UHVSHFW WR WKH VWRFN SULFH WLPHV WKH ULVN SUHPLXP

HDUQHG KROGLQJ WKH VWRFN� $V WKH VWRFN SULFH DSSURDFKHV ]HUR� WKLV HODVWLFLW\ JURZV ZLWKRXW ERXQG�

DQG VR GRHV WKH ULVN SUHPLXP RI WKH RSWLRQ� %HFDXVH RXU RSWLRQV DUH $PHULFDQ DQG GR QRW KDYH DQ

H[SLUDWLRQ GDWH� WKH ULVN SUHPLXP GRHV QRW JR WR LQnQLW\� EXW LV LQ IDFW ERXQGHG DERYH�

�� 8QNQRZQ 3UREDELOLW\ RI 6XFFHVV

,Q WKH JHQHUDO YHUVLRQ RI RXU PRGHO� WKH nUP OHDUQV DERXW LWV WHFKQLFDO FDSDELOLWLHV DV LW LQYHVWV

LQ 5'� $V ZH KDYH DOUHDG\ DUJXHG� DOWKRXJK WKH XQFHUWDLQW\ LQ WKLV SURFHVV LV LGLRV\QFUDWLF� LWV

UHVROXWLRQ DmHFWV WKH ULVN SUHPLXP�

7KH LPSRUWDQFH RI XQFHUWDLQW\ RYHU WHFKQLFDO DELOLW\ FDQ EH JDXJHG E\ FRQWUDVWLQJ WKH YDOXHV

LQ 7DEOH �� WKH YDOXH RI WKH SURMHFW DW GLmHUHQW VWDJHV RI GHYHORSPHQW� WR WKRVH LQ 7DEOH �� WKH

��

1XPEHU RI &RPSOHWHG 6WDJHV �Q�DQG )DLOXUHV �P�

p � � � ��

��

7DEOH �� 9DOXH RI WKH 5' 9HQWXUH ZKHQ { LV 8QNQRZQ DQG WKH QXPEHU RI VXFFHVVHTXDOV WKH QXPEHU RI IDLOXUHV� 7KH FHOOV LQ WKH WDEOH VKRZ WKH YDOXH RI D YHQWXUH WKDW UHTXLUHV ��VWDJHV WR FRPSOHWH IRU D JLYHQ p DQG QXPEHU RI FRPSOHWHG VWDJHV� ZLWK DQ HTXLYDOHQW DPRXQW RI IDLOXUHV �L�H��

P Q�� %ROG IDFH YDOXHV FRUUHVSRQG WR PRWKEDOOHG SURMHFWV �SURMHFWV WKDW KDYH DQ HTXDO QXPEHU RI VXFFHVV

DQG IDLOXUHV DUH QHYHU DEDQGRQHG IRU WKLV SDUDPHWHUL]DWLRQ��

FRPSDUDEOH WDEOH ZKHQ { LV NQRZQ� 7R IDFLOLWDWH WKLV FRPSDULVRQ� 7DEOH � RQO\ UHSRUWV YDOXHV ZKHQ

Q P� VLQFH ZKHQ WKH QXPEHU RI VXFFHVVHV HTXDOV WKH QXPEHU RI IDLOXUHV� x{�Q� Q� �� ZKLFK LV

WKH YDOXH IRU { LQ WKH FDVH ZKHQ { LV NQRZQ� 7KH H[WUD XQFHUWDLQW\ DGGV FRQVLGHUDEOH YDOXH WR WKH

SURMHFW LQ LWV HDUO\ VWDJHV� )RU H[DPSOH� EHIRUH DQ\ UHVHDUFK LV XQGHUWDNHQ� WKH SURMHFW LV DQ RUGHU RI

PDJQLWXGH PRUH YDOXDEOH ZKHQ { LV XQNQRZQ UDWKHU WKDQ NQRZQ� $ORQJ ZLWK LQFUHDVLQJ WKH YDOXH

RI WKH JURZWK RSWLRQ� WKH DGGHG XQFHUWDLQW\ DERXW WKH SUREDELOLW\ RI VXFFHVV DOVR DOWHUV WKH RSWLPDO

LQYHVWPHQW SROLF\� )RU H[DPSOH� DW WKH RXWVHW� ZKHQ p �� WKH 5 ' LV PRWKEDOOHG ZKHQ { LV

NQRZQ� EXW LV XQGHUWDNHQ ZKHQ LW LV XQNQRZQ� ,Q WKH ODWWHU FDVH� XQGHUWDNLQJ WKH UHVHDUFK DOORZV

WKH nUP WR REWDLQ D EHWWHU HVWLPDWH RI {� DQG WKH nUP KDV DQ RSWLRQ�OLNH FODLP RQ WKLV YDULDEOH� $

SRVLWLYH UHDOL]DWLRQ RI { LV YHU\ YDOXDEOH� VR WKH nUP DWWHPSWV WKH UHVHDUFK LQLWLDOO\ LQ WKH KRSHV

WKDW WKH WUXH { LV JUHDWHU WKDQ �� &XULRXVO\� DOWKRXJK LW LV LQLWLDOO\ RSWLPDO WR LQYHVW LQ 5'�

DIWHU IRXU VXFFHVVHV DQG IDLOXUHV LW LV RSWLPDO WR PRWKEDOO� HYHQ WKRXJK WKH SRVWHULRU H[SHFWDWLRQ

RI WKH WUXH SUREDELOLW\ RI VXFFHVV UHPDLQV DW �� 2I FRXUVH� DIWHU IRXU VXFFHVV DQG IDLOXUHV WKH

FRPSDQ\ KDV D PXFK PRUH SUHFLVH HVWLPDWH RI {� 7KH UHGXFHG XQFHUWDLQW\ DERXW { PDNHV WKH

VLWXDWLRQ PRUH OLNH WKH NQRZQ { FDVH� DQG WKHUH LW LV RSWLPDO WR PRWKEDOO ZLWK D �� FKDQFH RI

VXFFHVV� ,Q JHQHUDO WKH WDEOH VKRZV WKDW DV WKH QXPEHU RI VXFFHVVHV DQG IDLOXUHV LQFUHDVHV� WKH YDOXH

RI WKH SURMHFW DSSURDFKHV WKH YDOXH LQ WKH NQRZQ { FDVH��

7KH HmHFW RI WKH XQFHUWDLQW\ LQ { LV DOVR HYLGHQW LQ WKH SORW RI WKH ULVN SUHPLXP ZKHQ WKH

QXPEHU RI VXFFHVVHV HTXDOV WKH QXPEHU RI IDLOXUHV� �6HH )LJXUH �� :KHQ WKH QXPEHU RI VXFFHVV

DQG IDLOXUHV LV ODUJH� WKH SORW ORRNV YHU\ VLPLODU WR )LJXUH �� +RZHYHU� ZKHQ WKHUH KDYH EHHQ YHU\

IHZ VXFFHVV DQG IDLOXUHV� KROGLQJ FRQVWDQW WKH SRVWHULRU SUREDELOLW\ RI VXFFHVV� IRU KLJK YDOXHV RI p

WKH ULVN SUHPLXP FDQ LQFUHDVH DV WKH SURMHFW SURFHHGV� ,Q WKLV FDVH WKH GHFUHDVHG XQFHUWDLQW\ LQ {

UDLVHV pe� WKXV PDNLQJ LW PRUH OLNHO\ WKDW D p UHDOL]DWLRQ ZLOO RFFXU WKDW ZLOO PRWKEDOO WKH SURMHFW�

�7KH YDOXH LQ WKH XQNQRZQ FDVH FDQ DFWXDOO\ EH ORZHU ZKHQ Q LV ODUJH �H�J�� Q �� 7KLV UHVXOWV� SUHVXPDEO\�IURP WKH DV\PPHWU\ LQ WKH SRVWHULRU GLVWULEXWLRQ RI x{ LQGXFHG E\ WKH IDFW WKDW LQ WKH ODWWHU VWDJHV� 1 �� Pe�1��6R� IRU H[DPSOH� ZKHQ Q �� WKH DEVROXWH PRVW WKDW x{ FDQ GHSDUW IURP LWV FXUUHQW YDOXH LQ WKH SRVLWLYH GLUHFWLRQ LV��

�� 2Q WKH RWKHU KDQG WKH QHJDWLYH GHSDUWXUH FDQ EDVLFDOO\ JR DOO WKH ZD\ WR ��

�� VLQFH LW WDNHV

XS WR �� IDLOXUHV WR DEDQGRQ�

��

0

4

8

12

16

20

n and m

2.

4.

6.

8.

10.

12.

14.

Delta

10

15

20

4

8

12

16

20

n and m

2.

4.

6.

8.

10.

12.

14.

Delta

)LJXUH �� 5LVN 3UHPLXP RI WKH 5' 9HQWXUH ZKHQ { LV 8QNQRZQ DQG WKH QXPEHU RIVXFFHVV HTXDOV WKH QXPEHU RI IDLOXUHV� 7KH ULVN SUHPLXP �H[SUHVVHG LQ ��DQQXP� RI SURMHFW WKDW

UHTXLUHV �� VWDJHV WR FRPSOHWH LV SORWWHG DV D IXQFWLRQ RI p� �ODEHOHG ?'HOWD� LQ WKH SORW� DQG WKH QXPEHU

RI FRPSOHWHG VWDJHV �Q� DQG IDLOXUHV �P�� 7KH ULVN SUHPLXP LV GHnQHG WR EH WKH GLmHUHQFH EHWZHHQ WKH

FRQWLQXRXVO\ FRPSRXQGHG H[SHFWHG UHWXUQ RI WKH SURMHFW DQG WKH ULVNOHVV UDWH�

��

7KH HDUO\ KLVWRU\ LV VR LQIRUPDWLYH WKDW WKLV GHFUHDVHG XQFHUWDLQW\ VZDPSV WKH FRXQWHUYDLOLQJ HmHFW

WKDW WKH QXPEHU RI VWDJHV OHIW WR FRPSOHWH LV VPDOOHU�

1XPEHU RI &RPSOHWHG 6WDJHV RU 3DVW 6XFFHVVHV �Q��

7DEOH �� 1XPEHU RI )DLOXUHV 7KDW 7ULJJHU $EDQGRQPHQW� 7KH FHOOV LQ WKH WDEOH VKRZ� IRUD JLYHQ QXPEHU RI FRPSOHWHG VWDJHV �RU SDVW VXFFHVVHV�� WKH QXPEHU RI IDLOXUHV DW ZKLFK LW LV RSWLPDO WR

DEDQGRQ WKH �� VWDJH YHQWXUH�

7KH HmHFW RI HDUO\ OHDUQLQJ E\ GRLQJ LV HYHQ PRUH FOHDUO\ VHHQ LQ 7DEOH �� ZKLFK OLVWV WKH QXPEHU

RI IDLOXUHV WKDW WULJJHU WKH DEDQGRQPHQW GHFLVLRQ� (DUO\ LQ WKH OLIH RI DQ 5' SURMHFW WHFKQLFDO

IDLOXUH LV YHU\ FRVWO\� ,QLWLDOO\� ZLWK MXVW WZR IDLOHG DWWHPSWV DQG QR VXFFHVVHV LW LV RSWLPDO WR

DEDQGRQ WKH SURMHFW� ,I WKH WUXH SUREDELOLW\ RI VXFFHVV LV �� WKHQ �� RI WKH WLPH WKH SURMHFW

ZLOO EH DEDQGRQHG EHIRUH HYHQ RQH VWDJH LV FRPSOHWH� /DWHU LQ WKH OLIH RI DQ 5' SURMHFW� IDLOXUH

LV PXFK OHVV FRVWO\� :LWK �� VWDJHV FRPSOHWH� LW ZRXOG WDNH �� IDLOXUHV EHIRUH WKH DEDQGRQPHQW

RSWLRQ LV H[HUFLVHG� 7KH UHDVRQ IRU WKLV GRHV QRW GHULYH H[FOXVLYHO\ IURP WKH HmHFW RI SDVW VXFFHVV

RQ SRVWHULRUV� ,W DOVR UHoHFWV WKH IDFW WKDW WKH H[SHFWHG FRVW WR FRPSOHWLRQ LV ORZHU ZKHQ RQO\ D

IHZ VWDJHV UHPDLQ� 1HYHUWKHOHVV� WKHVH QXPEHUV HPSKDVL]H WKH LPSRUWDQW UROH OXFN SOD\V LQ ZKLFK

SURMHFWV nUPV XOWLPDWHO\ GHYHORS� 7ZR LGHQWLFDO nUPV GRLQJ 5' RQ LGHQWLFDO SURMHFWV FDQ� EHFDXVH

RI VPDOO EXW HDUO\ GLmHUHQFHV WKH UHVROXWLRQ RI WKH WHFKQRORJLFDO XQFHUWDLQW\� EH OHG WR FRPSOHWHO\

GLmHUHQW RXWFRPHV�

%HFDXVH WKH ULVN SUHPLXP LV YHU\ VHQVLWLYH DURXQG WKH PRWKEDOO SRLQW� )LJXUH � FDQ DOVR EH XVHG

WR DVVHVV WKH GLmHUHQFHV EHWZHHQ LQYHVWPHQW SROLFLHV ZKHQ { LV NQRZQ DQG ZKHQ LW LV XQNQRZQ� 7KH

UHJLRQ ZHUH WKH SURMHFW LV PRWKEDOOHG FRUUHVSRQGV WR WKH SODWHDX ZKHUH WKH ULVN SUHPLXP LV oDW DQG

FORVH WR WKH XSSHU ERXQG FDOFXODWHG LQ 3URSRVLWLRQ �� 7KH FULWLFDO YDOXHV WKDW WULJJHU VXVSHQVLRQ RI

LQYHVWPHQW FDQ EH GHGXFHG IURP WKH SRLQWV DW ZKLFK WKH SORW MXPSV XS WR WKLV SODWHDX� ,QLWLDOO\� WKLV

SRLQW RFFXUV DW pe�� { �� VHH )LJXUH �� +RZHYHU� ZKHQ { LV NQRZQ� WKLV RFFXUV DW pe�� { ��VHH )LJXUH �� $V ZH KDYH DOUHDG\ DUJXHG� WKLV GLmHUHQFH RFFXUV EHFDXVH ZKHQ { LV XQNQRZQ� RQH

UHDVRQ WR GHYHORS LV WR OHDUQLQJ DERXW \RXU WHFKQLFDO DELOLW\� $V WKLV XQFHUWDLQW\ LV UHVROYHG WKH

GLmHUHQFHV EHWZHHQ WKH RSWLPDO LQYHVWPHQW SROLFLHV GLVDSSHDU� 6R� IRU H[DPSOH� ZLWK Q P ��

LQ ERWK FDVHV pe LV FORVH WR ��7KH UDWLR RI VXFFHVVHV WR IDLOXUHV DOVR KDV DQ LPSRUWDQW HmHFW RQ WKH ULVN SUHPLXP� )LJXUH �

SORWV WKH ULVN SUHPLXP DV D IXQFWLRQ RI WKH FXUUHQW p DQG WKH QXPEHU RI SDVW IDLOXUHV DW IRXU

GLmHUHQW VWDJHV RI GHYHORSPHQW� )RU H[DPSOH� ZKHQ p �� LW LV RSWLPDO WR GHYHORS LQLWLDOO\ DQG VR

WKH ULVN SUHPLXP LV �� SHU \HDU �)LJXUH �� XSSHU OHIW SDQHO�� :LWK MXVW RQH IDLOXUH� WKH SURMHFW

LV PRWKEDOOHG DQG VR WKH ULVN SUHPLXP MXPSV WR WKH XSSHU ERXQG RI �� $UHDV EHORZ WKH oDW

SDUWV RI WKH SORW DUH UHJLRQV LQ ZKLFK WKH SURMHFW LV XQGHUWDNHQ DQG WKH HPSW\ DUHDV LQ WKH SORWV

FRUUHVSRQG WR YDOXHV IRU ZKLFK WKH YHQWXUH LV DEDQGRQHG�� 2Q WKH RWKHU KDQG ZKHQ �� VWDJHV DUH

��

FRPSOHWHG ZLWKRXW DQ\ IDLOXUHV �)LJXUH �� XSSHU ULJKW SDQHO�� IRU WKH VDPH YDOXH RI p� WKH ULVN

SUHPLXP LV RQO\ �� FORVH WR WKH ORZHU ERXQG� $ IDLOXUH DW WKLV SRLQW FKDQJHV WKH SUHPLXP RQO\

PDUJLQDOO\� WR �� 7KH VLWXDWLRQ LV TXLWH GLmHUHQW ZKHQ WKH QXPEHU RI SDVW IDLOXUHV LV ODUJH� ,I

WKHUH KDYH EHHQ �� SDVW IDLOXUHV� WKH ULVN SUHPLXP LV �� DQG LW MXPSV WR �� LI RQH PRUH

IDLOXUH LV UHDOL]HG� :LWK �� VWDJHV FRPSOHWH �)LJXUH �� ORZHU ULJKW SDQHO�� WKH SURMHFW LV VR IDU DORQJ

WKDW XQOHVV WKH QXPEHU RI IDLOXUHV LV YHU\ ODUJH �LQ H[FHVV RI ��_QRWH WKH VFDOH FKDQJH RQ WKH SORW�

WKH QXPEHU RI SDVW IDLOXUHV PDNHV YHU\ OLWWOH GLmHUHQFH DQG WKH ULVN SUHPLXP LV ZLWKLQ �� RI WKH

ORZHU ERXQG�

7KH FDVH ZLWK �� FRPSOHWHG VWDJHV LQ )LJXUH � �ORZHU OHIW SDQHO� SURYLGHV D QLFH LOOXVWUDWLRQ

RI WKH GLmHUHQFH EHWZHHQ WKH GHFLVLRQ WR PRWKEDOO DQG DEDQGRQ� 5HFDOO WKDW WKH UHVROXWLRQ RI

WHFKQRORJLFDO XQFHUWDLQW\ LV LGLRV\QFUDWLF� &RQVHTXHQWO\� WKH UHDVRQ WKH GHFLVLRQ WR PRWKEDOO DmHFWV

WKH ULVN SUHPLXP LV WKDW WKH RSWLPDO H[HUFLVH SROLF\ GHSHQGV RQ WKH FXUUHQW OHYHO RI p� 7KH GHFLVLRQ

WR DEDQGRQ� KRZHYHU� GHSHQGV RQO\ RQ WKH QXPEHU RI SDVW IDLOXUHV� VR WKLV RSWLRQ GRHV QRW FRPPDQG

D ULVN SUHPLXP� 7KLV FDQ EH FOHDUO\ GLVFHUQHG IURP WKH SORW� :KHQ p LV VPDOO� LW LV SRVVLEOH WKDW

ZLWK HQRXJK IDLOXUHV� WKH SURMHFW ZLOO EH PRWKEDOOHG� 6LQFH WKH PRWKEDOO RSWLRQ FRPPDQGV D ULVN

SUHPLXP� WKLV FRUUHVSRQGV WR WKH oDW SDUW RI WKH SORW� $V WKH QXPEHU RI IDLOXUHV LQFUHDVH WKH ULVN

SUHPLXP ULVHV GUDPDWLFDOO\ WR WKH XSSHU ERXQG� :KHQ p LV ODUJH� LW LV QHYHU RSWLPDO WR PRWKEDOO�

DOWKRXJK IRU D ODUJH HQRXJK QXPEHU RI IDLOXUHV� LW PLJKW EH RSWLPDO WR DEDQGRQ� +RZHYHU� VLQFH WKH

DEDQGRQPHQW RSWLRQ GRHV QRW FRPPDQG D ULVN SUHPLXP� WKH ULVN SUHPLXP LQFUHDVHV RQO\ VOLJKWO\�

DV WKH QXPEHU RI IDLOXUHV DSSURDFKHV Pe�� ,Q )LJXUH � ZH SORW H[SHFWHG FDVK oRZV RYHU WKH QH[W �� \HDUV RI D SURMHFW LQ LWV LQLWLDO VWDJH DV

D IXQFWLRQ RI WKH FXUUHQW OHYHO RI p� 7KH H[SHFWHG FDVK oRZV DUH FDOFXODWHG E\ DYHUDJLQJ WKH UHDOL]HG

FDVK oRZ LQ HDFK SHULRG RYHU �� VLPXODWLRQ SDWKV IRU HDFK LQLWLDO YDOXH RI p� :H JHQHUDWH

HDFK VLPXODWLRQ SDWK E\ VWDUWLQJ DW Q �� P �� DQG WKH JLYHQ p� :H WKHQ GUDZ YDOXHV IRU WKH

UDQGRP YDULDEOHV LQ HDFK SHULRG DQG LPSOHPHQW RSWLPDO LQYHVWPHQW SROLFLHV� )URP WKH nJXUH� WKH

OLIH F\FOH FKDUDFWHULVWLFV RI WKH W\SLFDO SURMHFW LV TXLWH FOHDU� 7KH FXUUHQW �RU nUVW� FDVK oRZ LV HLWKHU

QHJDWLYH RU ]HUR� GHSHQGLQJ RQ ZKHWKHU LW LV RSWLPDO WR GHYHORS� $IWHU WKDW H[SHFWHG FDVK oRZV

DUH QHJDWLYH� UHoHFWLQJ WKH VWULFWO\ SRVLWLYH SUREDELOLW\ WKDW WKH SURMHFW ZLOO EH LQ GHYHORSPHQW�

7KH\ DOVR EHFRPH OHVV QHJDWLYH WKH PRUH GLVWDQW WKH\ DUH LQ WLPH� UHoHFWLQJ WKH RSWLPDO GHFLVLRQ WR

PRWKEDOO XQHFRQRPLFDO SURMHFWV� DQG WKXV FXW H[SHFWHG GHYHORSPHQW FRVWV� 6LQFH WKH SRVLWLYH FDVK

oRZV DVVRFLDWHG ZLWK WKH FRPSOHWHG SURMHFW DUH ODUJH� LQ DEVROXWH YDOXH� FRPSDUHG WR WKH FRVWV�

RQFH WKH PLQLPXP nYH�\HDU GHYHORSPHQW WLPH KDV SDVVHG WKH H[SHFWHG FDVK oRZV TXLFNO\ EHFRPH

SRVLWLYH� 7KH LQFUHDVHV LQ WKH H[SHFWHG FDVK oRZ HYHQWXDOO\ VWRS DQG WKH\ JUDGXDOO\ GHFOLQH DV WKH

SUREDELOLW\ RI REVROHVFHQFH LQFUHDVHV JHRPHWULFDOO\��

�7KLV VPDOO LQFUHDVH UHVXOWV IURP WKH RSWLRQ WR PRWKEDOO� VLQFH pe��P� LV LQFUHDVLQJ LQ P� DV LV HYLGHQW IURP WKHSORW�

�$OWKRXJK QRW HYLGHQW LQ WKH SORW� WKHUH DUH FDVHV ZKHUH WKH H[SHFWHG FDVK oRZV LQFUHDVH DIWHU �� \HDUV� $ IHZRI WKH YHU\ EDG SURMHFWV DUH PRWKEDOOHG IRU ORQJ SHULRGV RI WLPH� 5HFDOO WKDW RQFH D SURMHFW LV PRWKEDOOHG� LW ZLOOQRW EH DEDQGRQHG EHFDXVH QR ?OHDUQLQJ� DERXW WHFKQLFDO SURGXFWLYLW\ WDNHV SODFH� 7KH G\QDPLFV RI WKH p�W� SURFHVV�KRZHYHU� PHDQ WKHUH LV DOZD\V D FKDQFH WKDW VXFK D SURMHFW ZLOO HYHQWXDOO\ FRPH EDFN RQ OLQH� 7KLV ZLOO KDSSHQSUHFLVHO\ ZKHQ p�W� LV H[WUHPHO\ ODUJH� OHDGLQJ WR YHU\ ODUJH� WKRXJK XQOLNHO\� FDVK oRZV�

��

n = 16

1020

3040

5060

7080

90100

110120

1810

m

2.3.4.

5.6.

7.8.

9.10.

11.12.

13.14.

15.

Delta

10

15

20

1020

3040

5060

7080

90100

110120

1810

m

n = 19

1020

3040

5060

7080

90100

110120

1810

m

2.3.4.

5.6.

7.8.

9.10.

11.12.

13.14.

15.

Delta

10

15

20

1020

3040

5060

7080

90100

110120

1810

m

n = 0

1020

3040

5060

7080

90100

110120

1810

m

2.3.4.

5.6.

7.8.

9.10.

11.12.

13.14.

15.

Delta

10

15

20

1020

3040

5060

7080

90100

110120

1810

m

n = 11

1020

3040

5060

7080

90100

110120

1810

m

2.3.4.

5.6.

7.8.

9.10.

11.12.

13.14.

15.

Delta

10

15

20

1020

3040

5060

7080

90100

110120

1810

m

)LJXUH �� 5LVN 3UHPLXP RI WKH 5' YHQWXUH ZKHQ { LV XQNQRZQ� )RU IRXU GLmHUHQW VWDJHVRI FRPSOHWLRQ �Q �� DQG �� WKH ULVN SUHPLXP RI D SURMHFW WKDW UHTXLUHV �� VWDJHV WR FRPSOHWH LV

SORWWHG DV D IXQFWLRQ RI p� �ODEHOHG ?'HOWD� LQ WKH SORW� DQG P� WKH QXPEHU RI SDVW IDLOXUHV� 1RWH WKH VFDOH

FKDQJH RQ WKH ?P� D[LV_P LV SORWWHG LQ VWHSV RI RQH IURP � WR �� DQG WKHQ LQ VWHSV RI �� WR ��

%ODQN DUHDV FRUUHVSRQG WR YDOXHV IRU ZKLFK WKH YHQWXUH LV DEDQGRQHG� 7KH ULVN SUHPLXP LV GHnQHG WR EH WKH

GLmHUHQFH EHWZHHQ WKH FRQWLQXRXVO\ FRPSRXQGHG H[SHFWHG UHWXUQ RI WKH SURMHFW DQG WKH ULVNOHVV UDWH�

��

6

8

10

12

14

16

Delta

10

20

30

40

Years

-2

-1

0

1

2

6

8

10

12

14

16

Delta

-2

-

0

)LJXUH �� ([SHFWHG &DVK )ORZ DW WKH ,QLWLDO 6WDJH RI WKH 5' 9HQWXUH ZKHQ { LV8QNQRZQ� 7KH H[SHFWHG FDVK oRZ IRU D JLYHQ LQLWLDO p LV FDOFXODWHG DV IROORZV� �� VDPSOH SDWKVRYHU �� SHULRGV �� \HDUV� DUH LQGHSHQGHQWO\ VLPXODWHG� 7KH H[SHFWHG FDVK oRZ LV WKHQ FDOFXODWHG E\

DYHUDJLQJ WKH FDVK oRZ LQ HDFK SHULRG RYHU DOO WKH VDPSOH SDWKV� 7KH SORW VKRZV WKLV DYHUDJH RYHU WKH nUVW

�� SHULRGV �� \HDUV�� 7KH HPSW\ EODFN ER[ PDUNV ]HUR RQ WKH SORW�

��

1XPEHU RI &RPSOHWHG 6WDJHV �Q�DQG )DLOXUHV �P� �Q P�

p � � ��

&RVW RI &DS� �� ,55 ��

([S� 5HW� ��

&RVW RI &DS� �� ,55 ��

([S� 5HW� ��

&RVW RI &DS� �� ,55 ��

([S� 5HW� ��

&RVW RI &DS� �� ,55 ��

([S� 5HW� ��

&RVW RI &DS� �� ,55 ��

([S� 5HW� ��

&RVW RI &DS� �� ,55 ��

([S� 5HW� ��

7DEOH �� ,55 DQG VLQJOH SHULRG �RQH TXDUWHU� H[SHFWHG UHWXUQ RI WKH 5' 9HQWXUHZKHQ { LV XQNQRZQ� 7KH WDEOH VKRZV WKH FRVW RI FDSLWDO� ,55 DQG H[SHFWHG UHWXUQ RYHU WKH QH[W

TXDUWHU RI D YHQWXUH WKDW UHTXLUHV �� VWDJHV WR FRPSOHWH DW GLmHUHQW VWDJHV RI GHYHORSPHQW ZKHQ Q P�

7KH YDOXHV DUH H[SUHVVHG RQ DQ DQQXDO EDVLV LQ SHUFHQW IRU GLmHUHQW YDOXHV RI p� 7KH ,55 �FRVW RI FDSLWDO�

LV FDOFXODWHG IRU HDFK YDOXH RI p E\ VLPXODWLQJ WKH FDVK oRZV RYHU �� VDPSOH SDWKV DQG WKHQ nQGLQJ

WKH GLVFRXQW UDWH WKDW VHWV WKH SUHVHQW YDOXH RI WKH DYHUDJH FDVK oRZV HTXDO WR WKH FXUUHQW QHW LQYHVWPHQW

�YDOXH�� %ROG IDFH YDOXHV LQGLFDWHG PRWKEDOOHG SURMHFWV

��

,Q RXU PRGHO WKH H[SHFWHG UHWXUQ DVVRFLDWHG ZLWK EX\LQJ WKH SURMHFW DW LWV PDUNHW YDOXH DQG

KROGLQJ LW IRU RQH SHULRG FKDQJHV RYHU WKH SURMHFWV OLIH� 0RUH ZLGHO\ HPSOR\HG FDSLWDO EXGJHWLQJ

SURFHGXUHV W\SLFDOO\ H[SUHVV WKH SURMHFWV SURnWDELOLW\ DQG WKH FRVW RI IXQGV LQ WHUPV RI FRQVWDQW

GLVFRXQW UDWHV WKDW DSSO\ WR DOO H[SHFWHG FDVK oRZV LQ WKH IXWXUH� $FFRUGLQJO\� LQ 7DEOH � ZH XVH

WKH VLPXODWHG H[SHFWHG FDVK oRZV WR FDOFXODWH WKH LQWHUQDO UDWH RI UHWXUQ �,55� DQG D FRQVWDQW

?FRVW RI FDSLWDO� IRU WKH SURMHFW IRU D UDQJH RI YDOXHV RI p� DQG DW VHYHUDO VWDJHV RI GHYHORSPHQW�

:H IRFXV RQ FDVHV ZKHUH Q P� VR WKDW SRVWHULRU SUREDELOLWLHV RI VXFFHVV DUH FRQVWDQW� 7KH ,55 LV

WKH GLVFRXQW UDWH WKDW VHWV WKH SUHVHQW YDOXH RI DOO WKH SURMHFWV H[SHFWHG SK\VLFDO FDVK oRZV HTXDO

WR ]HUR� 7KH FRVW RI FDSLWDO VHWV WKH SUHVHQW YDOXH RI WKH SURMHFWV H[SHFWHG FDVK oRZV HTXDO WR WKH

SURMHFWV YDOXH� �7KXV� WKH FRVW RI FDSLWDO LV DOZD\V OHVV WKDQ WKH ,55 KHUH EHFDXVH WKH SURMHFWV

YDOXH LV SRVLWLYH�� 7KH WDEOH DOVR UHSRUWV WKH SURMHFWV H[SHFWHG UHWXUQ IRU FRPSDULVRQ� 1RWH WKDW

WKH H[SHFWHG FDVK oRZV UHoHFW RSWLPDO SROLFLHV�

)URP WKH WDEOH ZH VHH WKDW WKH ,55 IRU WKH SURMHFW LQFUHDVHV ZLWK SURJUHVV WRZDUGV FRPSOHWLRQ�

DV RQH ZRXOG H[SHFW RI D PHDVXUH RI WKH SURMHFWV DWWUDFWLYHQHVV� 7KH FRQVWDQW FRVW RI FDSLWDO�

KRZHYHU� LV QRQ�PRQRWRQLF� DQG ULVHV RYHU WKH HDUO\ VWDJHV RI GHYHORSPHQW� 7KLV DSSHDUV WR EH GXH

WR WKH GHFOLQH LQ XQFHUWDLQW\ DERXW WKH YDOXH RI {� ZKLFK� E\ GHFUHDVLQJ WKH YDOXH RI WKH UHDO RSWLRQV

HPEHGGHG LQ WKH SURMHFW� ORZHUV WKH SURMHFWV YDOXH GLVSURSRUWLRQDWHO\ UHODWLYH WR WKH H[SHFWHG FDVK

oRZV� 1RWH DOVR WKDW WKH ,55 RI WKH SURMHFW VRPHWLPHV OLHV EHORZ LWV H[SHFWHG UHWXUQ HYHQ LQ WKH

DUHD ZKHUH LW LV RSWLPDO WR PDNH WKH LQYHVWPHQW �H�J�� LQ WKH FHOO p �� Q P �� 7KLV LV QRW

VXUSULVLQJ� DV WKH ,55 LV D PHDVXUH RI DYHUDJH UHWXUQ HDUQHG RYHU WKH OLIH RI WKH SURMHFW� ZKLOH WKH

H[SHFWHG UHWXUQ DSSOLHV WR DQ LQYHVWPHQW LQ WKH SURMHFW IRU WKH FXUUHQW SHULRG�

�� 'LVWULEXWLRQ RI 5HDOL]HG 5HWXUQV

:H QRZ WXUQ WR WKH GLVWULEXWLRQ RI UHDOL]HG UHWXUQV� :H nUVW GHULYH WKLV GLVWULEXWLRQ� DQDO\WLFDOO\�

IRU WKH FDVH ZLWK QR n[HG FRVWV ZKHUH ZH KDYH D FORVHG�IRUP VROXWLRQ IRU WKH YDOXH� :H WKHQ XVH

QXPHULFDO PHWKRGV WR GHULYH WKH GLVWULEXWLRQ LQ WKH JHQHUDO FDVH�

:H FDQ VKRZ DQDO\WLFDOO\ IRU WKH VLPSOHVW FDVH WKDW WKH UHDOL]HG UHWXUQV DUH ELPRGDO�

3URSRVLWLRQ �� :KHQ WKHUH DUH QR n[HG 5' FRVWV �D �� WKH GLVWULEXWLRQ RI WKH VLQJOH�SHULRG

FRQWLQXRXVO\ FRPSRXQGHG UHWXUQ� FRQGLWLRQDO RQ VXUYLYDO� RI D SURMHFW ZLWK Q � 1 VWDJHV FRPSOHWHG

DQG P � Pe�Q� IDLOXUHV LV JLYHQ E\ D PL[WXUH RI WZR 1RUPDO GLVWULEXWLRQV ZLWK PHDQV .�Q�P� �

ORJ�K�Q � ��P�� DQG .�Q�P� � ORJ�K�Q�P � �� WKH VDPH VWDQGDUG GHYLDWLRQ �}p� DQG PL[WXUH

SUREDELOLW\ x{�Q�P� IRU WKH IRUPHU GLVWULEXWLRQ DQG �b x{�Q�P� IRU WKH ODWWHU� ZKHUH�

.�Q�P� s xb }�

�b ORJ �E� K�Q�P��

7R XQGHUVWDQG ZK\ WKH GLVWULEXWLRQ KDV WKLV IRUP� UHFDOO WKDW LQ WKLV FDVH LW LV QHYHU RSWLPDO

WR PRWKEDOO WKH SURMHFW� 6LQFH WKH UHWXUQ ZKHQ WKH SURMHFW LV DEDQGRQHG LV XQLQWHUHVWLQJ� ZH FDQ

��

OLPLW DWWHQWLRQ WR FDVHV LQ ZKLFK GHYHORSPHQW LV EHLQJ XQGHUWDNHQ� 7KHUH DUH WZR GLmHUHQW NLQGV RI

XQFHUWDLQW\ WKDW DmHFW YDOXH� )LUVW WKHUH LV WKH XQFHUWDLQW\ LQ p�W�� ZKLFK LV QRUPDOO\ GLVWULEXWHG�

7KHQ WKHUH LV WKH XQFHUWDLQW\ DVVRFLDWHG ZLWK WKH WHFKQLFDO ULVN� 6LQFH D VWDJH LV HLWKHU FRPSOHWHG

RU QRW� WKLV XQFHUWDLQW\ KDV D GLVFUHWH GLVWULEXWLRQ� 7KH PL[WXUH UHVXOWV IURP WKH FRPELQDWLRQ RI

WKHVH WZR HmHFWV�

7R H[DPLQH WKH UHDOL]HG UHWXUQV LQ WKH JHQHUDO FDVH� ZH VLPXODWH WKH VLQJOH SHULRG UHDOL]HG UHWXUQ

RI WKH SURMHFW DQDO\]HG LQ WKH SUHYLRXV VHFWLRQ DW GLmHUHQW OHYHOV RI p�W�� )LJXUH � DUH WKH UHVXOWLQJ

KLVWRJUDPV� :KDW LV SDUWLFXODUO\ VWULNLQJ DERXW WKH nJXUH LV KRZ YDULDEOH UHDOL]HG UHWXUQV DUH _

WKH SORWV VKRZ WKDW PRQWKO\ UHDOL]HG UHWXUQV RI �� RU �� DUH TXLWH OLNHO\�

:H KDYH VKRZQ WKDW DV p LQFUHDVHV� YDOXHV DQG SROLFLHV ZLWK SRVLWLYH n[HG FRVWV FRQYHUJH WR WKH

QR�n[HG�FRVWV FDVH� ,W LV WKHUHIRUH QRW VXUSULVLQJ WKDW ZKHQ p�W� LV ODUJH WKH UHVXOWLQJ GLVWULEXWLRQ LV

D PL[WXUH RI WZR GLVWULEXWLRQV� 6LQFH WKH SUREDELOLW\ RI VXFFHVV LV IXQFWLRQ RI SDVW SHUIRUPDQFH� WKH

PL[WXUH SUREDELOLWLHV �UHODWLYH ZHLJKWV� RI WKH WZR GLVWULEXWLRQV GHSHQGV RQ WKH VWDJH RI FRPSOHWLRQ�

7KH IRUP RI WKH GLVWULEXWLRQ RI UHDOL]HG UHWXUQV FKDQJHV LQ D GLVFRQWLQXRXV ZD\ DW p pe�� DSSUR[LPDWHO\ �� IRU RXU SDUDPHWHUL]DWLRQ�� )RU YDOXHV RI p�W� DERYH WKLV WKH UHVHDUFK LV XQ�

GHUWDNHQ DQG WKH IRUP RI WKH GLVWULEXWLRQ PLUURUV WKH QR�n[HG�FRVWV FDVH� :KHQ p�W� LV EHORZ WKLV

QXPEHU WKH GLVWULEXWLRQ LV TXDOLWDWLYHO\ GLmHUHQW� 7KLV FRUUHVSRQGV WR WKH FDVH ZKHQ WKH SURMHFW LV

PRWKEDOOHG� 6LQFH QR WHFKQLFDO XQFHUWDLQW\ LV UHVROYHG IRU PRWKEDOOHG SURMHFWV� RQH PLJKW UHDVRQ

WKDW WKH GLVWULEXWLRQ ZLOO QR ORQJHU EH ELPRGDO� ,Q IDFW� DV WKH FDVH ZKHQ p�W� �� LQ )LJXUH

� VKRZV� HYHQ ZKHQ WKH SURMHFW LV PRWKEDOOHG� WKH GLVWULEXWLRQ RI UHWXUQV UHVXOWV IURP D PL[WXUH

RI WZR GLVWULEXWLRQV� 2QH SRUWLRQ RI WKH GLVWULEXWLRQ FRUUHVSRQGV WR UDQJHV RI p�W � �� ZKHUH WKH

SURMHFW UHPDLQV PRWKEDOOHG DQG WKH RWKHU FRUUHVSRQGV WR WKH UDQJHV RI p�W� �� ZKHUH WKH SURMHFW

ZLOO EH XQGHUWDNHQ�

)LJXUH � FRQWDLQV KLVWRJUDPV RI UHDOL]HG UHWXUQV RI D SURMHFW DW GLmHUHQW VWDJHV RI GHYHORSPHQW

IRU p �� $W WKLV OHYHO RI p� WKH SURMHFW LV XQGHU DFWLYH GHYHORSPHQW IRU Q P � �XSSHU OHIW

SDQHO�� %\ FRPSDULQJ WKLV SDQHO WR WKH ORZHU ULJKW SDQHO LQ )LJXUH �� WKH HmHFW RI WKH LQIRUPDWLRQ

UHYHODWLRQ DVVRFLDWHG ZLWK WKH QXPEHU RI FRPSOHWHG VWDJHV �IDLOXUHV� LV HYLGHQW� ,Q ERWK FDVHV WKH

SURMHFW LV XQGHU GHYHORSPHQW� KRZHYHU LQ WKH SDQHO LQ )LJXUH � UHWXUQV DUH PXFK PRUH GLmXVH�

UHoHFWLQJ WKH KLJKHU OHYHO RI XQFHUWDLQW\ �DERXW {�� ,Q ERWK FDVHV D IDLOXUH DW WKLV SRLQW ZLOO OLNHO\

FDXVH WKH SURMHFW WR EH PRWKEDOOHG� +RZHYHU� LQ WKH FDVH GHSLFWHG LQ )LJXUH �� EHFDXVH RI WKH

GHFUHDVHG XQFHUWDLQW\ DVVRFLDWHG ZLWK WKH QXPEHU RI SDVW IDLOXUHV� WKHUH LV VRPH OLNHOLKRRG RI D

JRRG HQRXJK UHVROXWLRQ LQ p WKDW HYHQ RQH PRUH IDLOXUH FDQ EH WROHUDWHG ZLWKRXW PRWKEDOOLQJ WKH

SURMHFW� 6LPLODUO\� WKHUH LV VRPH OLNHOLKRRG RI D EDG HQRXJK UHVROXWLRQ LQ p WKDW HYHQ D SURMHFW WKDW

VXFFHVVIXOO\ FRPSOHWHV DQRWKHU VWDJH FRXOG EH PRWKEDOOHG� 7KXV WKH WZR GLVWULEXWLRQV WKDW IRUP

WKH PL[WXUH LQ XSSHU OHIW SDQHO RI )LJXUH � ?RYHUODS��

:KHQ WKH SURMHFW LV HYHQ IXUWKHU DORQJ� Q ��P � �ORZHU OHIW SDQHO RI )LJXUH �� RU

Q P �� ORZHU ULJKW SDQHO RI )LJXUH �� WKHUH LV QR UHDOLVWLF SUREDELOLW\ WKDW RYHU WKH QH[W

SHULRG WKH SURMHFW ZLOO EH PRWKEDOOHG� VR WKH GLVWULEXWLRQ LV VLQJOH SHDNHG� 7KH XSSHU ULJKW SDQHO�

��

-6. -4. -2. 0. 2.

∆HtL=9.540

-6. -4. -2. 0. 2.

∆HtL=16

-6. -4. -2. 0. 2.

∆HtL=4

-6. -4. -2. 0. 2.

∆HtL=9.539

)LJXUH �� 5HDOL]HG 5HWXUQ RI WKH 5' YHQWXUH �FRQGLWLRQDO RQ VXUYLYDO� ZKHQ { LVXQNQRZQ� )RU D SURMHFW DW LWV LQLWLDO VWDJH �Q P �� WKH VLQJOH SHULRG FRQWLQXRXVO\ FRPSRXQGHG

UHWXUQ RI D SURMHFW LV FDOFXODWHG RYHU �� UHSHWLWLRQV DW IRXU GLmHUHQW YDOXHV RI p�W�� 7KH nJXUHV VKRZ

WKH UHVXOWLQJ GLVWULEXWLRQ RI UHWXUQV DV D IXQFWLRQ RI WKH FXUUHQW p�W�� JLYHQ WKDW WKH SURMHFW VXUYLYHG �L�H��

[�W� ��

��

-2. 0. 2.

n=10 m=5

-2. 0. 2.

n=m=10

-2. 0. 2.

n=m=5

-2. 0. 2.

n=5 m=10

)LJXUH �� 5HDOL]HG 5HWXUQ RI WKH 5' YHQWXUH �FRQGLWLRQDO RQ VXUYLYDO� ZKHQ { LVXQNQRZQ DQG p �� )RU D SURMHFW DW IRXU GLmHUHQW VWDJHV LQ LWV GHYHORSPHQW WKH VLQJOH SHULRG FRQ�WLQXRXVO\ FRPSRXQGHG UHWXUQ RI D SURMHFW LV FDOFXODWHG RYHU �� UHSHWLWLRQV DW IRXU GLmHUHQW VWDJHV RI

GHYHORSPHQW� 7KH nJXUHV VKRZ WKH UHVXOWLQJ GLVWULEXWLRQ RI UHWXUQV DV D IXQFWLRQ RI WKH FXUUHQW p�W�� JLYHQ

WKDW WKH SURMHFW VXUYLYHG �L�H�� [�W� ��

��

Q ��P �� VKRZV ZKDW KDSSHQV WR WKH SURMHFW LI LW GRHV QRW SURJUHVV IXUWKHU� ,Q WKLV FDVH� ZLWK

nYH PRUH IDLOXUHV� WKH RSWLPDO SROLF\ LV WR PRWKEDOO WKH SURMHFW� :LWK p �� WKHUH LV QR UHDOLVWLF

SUREDELOLW\ WKDW SRVLWLYH UHVROXWLRQ LQ p XQFHUWDLQW\ LQ WKH QH[W SHULRG FDQ FKDQJH WKLV� VR DJDLQ

WKH GLVWULEXWLRQ LV VLQJOHG SHDNHG DQG UHoHFWV WKH GLVWULEXWLRQ RI UHWXUQV RI D PRWKEDOOHG SURMHFW�

7KH GLVWULEXWLRQ RI UHDOL]HG UHWXUQV DOVR LOOXVWUDWHV QLFHO\ WKH VHSDUDWH UROHV RI V\VWHPDWLF DQG

XQV\VWHPDWLF ULVN� 7KH YDULDQFH RI UHWXUQV ZKHQ WKH SURMHFW LV PRWKEDOOHG LV FOHDUO\ FRQVLGHUDEO\

VPDOOHU WKDQ ZKHQ LW LV XQGHU GHYHORSPHQW DQG WKH ELPRGDOLW\ LV VR H[WUHPH �H�J�� FRPSDUH WKH WZR

WRS SDQHOV RI )LJXUH �� <HW WKH ULVN SUHPLXP GXULQJ WKH PRWKEDOO VWDJH LV PXFK ODUJHU WKDQ WKH

GHYHORSPHQW VWDJH� 7KH UHDVRQ IRU WKLV GLVSDULW\ LV WKDW WKH H[WUD YDULDQFH LQ WKH UHDOL]HG UHWXUQ

RI SURMHFWV XQGHU GHYHORSPHQW GHULYHV GLUHFWO\ IURP WKH LQFUHDVH LQ WHFKQRORJLFDO ULVN� 7KLV ULVN LV

LGLRV\QFUDWLF� VR LW GRHV QRW DmHFW WKH ULVN SUHPLXP�

� &RQFOXVLRQ

5' DQG VLPLODU LQYHVWPHQW SURMHFWV KDYH WKH SURSHUW\ WKDW PXFK RI WKH YDOXH RI WKH LQYHVWPHQW

LV DVVRFLDWHG ZLWK IXWXUH FDVK oRZV WKDW DUH FRQWLQJHQW RQ LQWHUPHGLDWH GHFLVLRQV� 7KH XQFHUWDLQW\

WKHVH SURMHFWV LQYROYH LV RI WZR GLVWLQFW W\SHV� 7KHUH LV SXUHO\ LGLRV\QFUDWLF ULVN DVVRFLDWHG ZLWK WKH

UHVROXWLRQ RI WHFKQLFDO XQFHUWDLQW\� 7KHUH LV DOVR ULVN DVVRFLDWHG ZLWK FDVK oRZV DIWHU GHYHORSPHQW

LV FRPSOHWH WKDW ZLOO KDYH DQ LPSRUWDQW V\VWHPDWLF FRPSRQHQW�

2XU DQDO\VLV KLJKOLJKWV WKH LPSRUWDQFH RI WKH QDWXUH RI WKH LQIRUPDWLRQ GHFLVLRQ PDNHUV FRQ�

GLWLRQ RQ ZKHQ WKH\ PDNH WKH LQWHUPHGLDWH LQYHVWPHQW GHFLVLRQV WR FRQWLQXH� H[SDQG� FRQWUDFW�

RU DEDQGRQ WKH SURMHFW� :KHQHYHU WKDW LQIRUPDWLRQ LQFOXGHV YDULDEOHV� VXFK DV IRUHFDVWV RI SRVW�

GHYHORSPHQW FDVK oRZV� WKDW KDYH D V\VWHPDWLF FRPSRQHQW� WKLV ZLOO LPSDUW FRQVLGHUDEOH V\VWHPDWLF

ULVN WR WKH SURMHFW� HYHQ ZKHQ WKH GHYHORSPHQW SURFHVV LWVHOI LQYROYHV RQO\ WHFKQLFDO ULVN� 2XU UH�

VXOWV VKRZ WKDW WKH V\VWHPDWLF ULVN� DQG WKH UHTXLUHG ULVN SUHPLXP� RI WKH YHQWXUH DUH KLJKHVW HDUO\

LQ LWV OLIH� DQG GHFUHDVH DV LW DSSURDFKHV FRPSOHWLRQ� GHVSLWH WKH LGLRV\QFUDWLF QDWXUH RI WKH WHFKQLFDO

ULVN� 7KH RSWLRQ�OLNH FKDUDFWHULVWLFV RI WKH YHQWXUH DOVR GHWHUPLQH WKH QDWXUH RI WKH GHSHQGHQFH

RI LWV YDOXH RQ WKH XQGHUO\LQJ VWDWH YDULDEOHV� $W FHUWDLQ WLPHV GXULQJ WKH SURMHFWV OLIH� LWV YDOXH

LV H[WUHPHO\ UHVSRQVLYH WR LQIRUPDWLRQ DERXW WHFKQLFDO SURJUHVV RU SRWHQWLDO SURnWDELOLW\� DQG DW

RWKHUV PXFK OHVV VR�

��

$SSHQGL[

$ 3URRIV

'HULYDWLRQ RI WKH 9DOXH )XQFWLRQ

,Q HTXDWLRQ �� QRWH WKDW WKH IXQFWLRQ 9 �c� DSSHDUV RQ ERWK VLGHV� DQG 9 �c� LV GHnQHG DV DVROXWLRQ WR D IXQFWLRQDO HTXDWLRQ� %HORZ ZH nUVW VKRZ WKDW WKH IXQFWLRQDO HTXDWLRQ LPSOLFLW LQ ��

KDV D XQLTXH VROXWLRQ� 7KHQ� ZH VKRZ WKDW WKLV VROXWLRQ LQGHHG SURYLGHV WKH YDOXH RI WKH 5'

SURMHFW FRQVLGHUHG LQ WKH SDSHU�

'HnQH = s 5� d 1� ZKHUH 1� s I�� J� /HW &�=� GHQRWH WKH VSDFH RI FRQWLQXRXVIXQFWLRQV GHnQHG RQ = ZKLFK LV D FRPSOHWH DQG VHSDUDEOH PHWULF VSDFH� /HW $ �

�bn DQG OHW 'GHQRWH D VXEVHW RI &�=� GHnQHG E\

' s�I � &

nnnn VXS�]�P��5�d1�

� MI�]�P�M]

w� $

��

' LV D FORVHG VXEVHW RI &�=� DQG LV� WKHUHIRUH� LWVHOI D FRPSOHWH� DQG VHSDUDEOH PHWULF VSDFH�)RU Q �� 1 b �� FRQVLGHU D IDPLO\ RI PDSSLQJV $Q � ' � & GHnQHG UHFXUVLYHO\ E\�

�$QI��]�P� PD[>�%QI��]�P�� &I��]�P�@ ZKHUH �%QI��]�P� HTXDOV

�x{�Q�P�

=9Q��] H[S�\��P�Q�\�G\ � �>�b x{�Q�P�@

=I�] H[S�\��P� ��Q�\�G\ b Db E]�

DQG

�&I��]�P� �

=I�] H[S�\��P�Q�\�G\�

9Q��c� GHQRWHV WKH XQLTXH n[HG SRLQW RI $Q�� ZKRVH H[LVWHQFH LV VKRZQ EHORZ� DQG ZKHUH

91 �]�P� s $]� 0RUHRYHU� Q�\� GHQRWHV WKH QRUPDO GHQVLW\ ZLWK PHDQ x b w b ��}� DQG YDUL�

DQFH }� DQG � {\ H[S�bU� � ��

%HJLQQLQJ ZLWK Q 1 b �� LW FDQ EH VKRZQ UHFXUVLYHO\ IRU Q �� 1 b �� WKDW �L� LII�c� � '� WKHQ ERWK %QI DQG &I DUH LQ '� 7KXV� $Q�c� PDSV ' LQWR ' IRU DOO Q �� 1 b ��LL� IRU DOO Q� $Q LV PRQRWRQLF �L�H� $QI w $QJ ZKHQHYHU I w J�� DQG �LLL� ZLWK �=�]�P� s � IRU

�]�P� � = DQG m w ��

$Q�I � m�=� � PD[>%QI � �m�= � &I � �m�= @ $QI � �m�= �

7KHQ� E\ %ODFNZHOOV n[HG SRLQW WKHRUHP �'XpH �� S� �� $Q LV D FRQWUDFWLRQ IRU � � Q �1 b � �VLQFH � � �� DQG KDV D XQLTXH n[HG SRLQW� 'HQRWH WKH n[HG SRLQW RI $Q E\ 9Q�p�P�� 7KLV

SURYHV WKH H[LVWHQFH RI D XQLTXH VROXWLRQ WR WKH IXQFWLRQDO HTXDWLRQ LPSOLFLW LQ �� 7R PDNH FOHDU

WKH GHSHQGHQFH RI WKH YDOXH RI WKH 5' SURMHFW DW GLmHUHQW VWDJHV RQ Q�W�� ZH GHQRWH WKH XQLTXH

n[HG SRLQW RI $Q� 9Q�p�P�� DOVR DV 9 �Q�P� p��

��

'HnQH : >W@ WR EH WKH YDOXH RI WKH 5' SURMHFW DW GDWH W� :KHQ [�W� �� : >W@ �� 7KXV� RQO\

WKH FDVH ZKHQ [�W� � QHHGV WR EH DQDO\]HG IXUWKHU� :H QRZ VKRZ WKDW 9 �Q�P� p� LV LQGHHG WKH

YDOXH DW GDWH W� �ZKHQ p�W� p� Q�W� Q P�W� P DQG [�W� �� RI WKH 5' SURMHFW FRQVLGHUHG LQ

WKLV SDSHU� 7R VHH WKLV� FRQVLGHU WKH SUREOHP RI YDOXLQJ DQ 5' SURMHFW WKDW LV VLPLODU LQ DOO UHVSHFWV

WR WKH RQH GHVFULEHG LQ 6HFWLRQ � H[FHSW WKDW WKH SURMHFW LV nQLWHO\ OLYHG� 7KDW LV� LI WKH 5' SURMHFW

KDV QRW EHHQ nQLVKHG E\ WLPH 7 � LW EHFRPHV ZRUWKOHVV DIWHU 7 � 7KH YDOXH RI VXFK DQ RSSRUWXQLW\

DW DQ\ W� GHQRWHG : �W� 7 �� FDQ EH FDOFXODWHG E\ EDFNZDUG UHFXUVLRQ� (YLGHQWO\� : �W� 7 � � IRU

DOO W ! 7 � $OVR� : �W� 7 � � IRU DQ\ W � 7 LI [�W� �� 0RUHRYHU� E\ EDFNZDUG UHFXUVLRQ VWDUWLQJ

DW 7 b �� IRU DQ\ W � 7 � : �W� 7 � HTXDOV -Q�p�W��P�W�� W� 7 � LI Q�W� Q DQG [�W� �� ZKHUH WKH

IXQFWLRQV -Q�c� W� 7 �� IRU Q �� 1 b �� DUH GHnQHG E\ -Q�c� W� 7 � PD[>.Q�c� W� 7 �� /Q�c� W� 7 �@ZLWK .Q�[�P� W� 7 � JLYHQ E\

�x{�Q�P�

=-Q��[ H[S�\��P� W�� 7 �Q�\�G\��>�bx{�Q�P�@

=-Q�[ H[S�\��P�� W�� 7 �Q�\�G\bDbE[

DQG

/Q�[�P� W� 7 �

=-Q�[ H[S�\��P� W� �� 7 �Q�\�G\�

VXEMHFW WR WKH ERXQGDU\ FRQGLWLRQV WKDW -1 �p�P� W� 7 � s $p IRU DOO W � 7 DQG -Q�p�P� 7� 7 � s �

IRU Q � 1 b ��:H GHnQH WKH YDOXH RI DQ LQnQLWHO\�OLYHG 5' RSSRUWXQLW\ WR EH WKH OLPLW RI WKH YDOXH RI

nQLWHO\�OLYHG RSSRUWXQLWLHV DV WKH OLIH RI WKH RSSRUWXQLW\ JRHV WR LQnQLW\� $FFRUGLQJO\� ZH VHW WKH

YDOXH RI WKH LQnQLWHO\�OLYHG 5' SURMHFW FRQVLGHUHG LQ RXU PRGHO WR EH OLP7��: �W� 7 ��

6LQFH -Q�c� W� 7 � LV PRQRWRQLFDOO\ LQFUHDVLQJ LQ 7 DQG -Q�p�P� W� 7 � � $p� IRU DOO 7 w �� WKH OLPLWRI -Q�c� W� 7 � DV 7 � � H[LVWV� 'HQRWH WKLV OLPLW DV -Q�c�� 8VLQJ WKH PRQRWRQH FRQYHUJHQFHWKHRUHP� LW FDQ WKHQ EH VKRZQ WKDW -Q�� LV D n[HG SRLQW RI $Q� 6LQFH WKHUH LV RQO\ RQH n[HG

SRLQW RI $Q� Q �� 1 b �� WKH YDOXH RI WKH LQnQLWHO\�OLYHG 5' SURMHFW DW DQ\ W HTXDOV�

OLP7��

: �W� 7 � -Q�W��p�W��P�W�� W�� 9 �Q�W��P�W�� p�W��

ZKHQ [�W� ��

3URRI RI 3URSRVLWLRQ �

$ VLQJOH WHUP LQ �� FDQ EH ZULWWHQ�

(W

|]�W� V�

]�W�p�W� V�[�W� V�

} p�W� >{\ H[S�xb U b w�@V ��

7KLV IROORZV IURP �L� ZULWLQJ ]�W�V� DQG p�W�V� LQ WHUPV RI ]�W� DQG p�W�� LL� FRPELQLQJ H[SRQHQWLDOV�

�LL� DSSO\LQJ WKH IRUPXOD IRU WKH PHDQ RI D ORJQRUPDO� DQG �LY� UHFRJQL]LQJ WKDW WKH RQO\ QRQ�]HUR

��

FRYDULDQFH EHWZHHQ WKH LQQRYDWLRQV LQ ]�W� DQG p�W� LV FRQWHPSRUDQHRXV DQG FRQVWDQW� ,Q DGGLWLRQ�

[�W� V� LV LQGHSHQGHQW RI ]�W� V� DQG p�W� V�� DQG (WI[�W� V�J {V\� 7KH IRUPXOD IRU DQ LQnQLWH

JHRPHWULF VHULHV WKHQ \LHOGV ��


7KH YDOXH RI WKH nUP JLYHQ LW KDV SDVVHG DOO 1 VWDJHV JLYHQ E\ 3URSRVLWLRQ �� p�W� ��b n�� LV D

FXP�GLYLGHQG YDOXH� VR�

ORJ(W>� �51W��@ ORJ

�#(W

K��bn p�W� ��[�W� ��

Lp�W�

K��bn b �

L�$ ��

6LQFH (W>[�W� ��@ {\ DQG (W>p�W� ��@ p�W�Hx� WKLV H[SHFWHG UHWXUQ VLPSOLnHV GLUHFWO\ WR

ORJ(W>� �51W��@ ORJ

t{\H

x

n

u U � w� ��


:H PXVW VKRZ WKDW 9 �Q�P� p� LV PRQRWRQH DQG FRQYH[ LQ p� IRU DOO n[HG P DQG Q� :H nUVW

GHPRQVWUDWH FRQYH[LW\�

6LQFH 9 �Q� c� LV WKH n[HG SRLQW RI $Q� DQG $Q LV D FRQWUDFWLRQ� LW LV VXpFLHQW WR VKRZ WKDW IRU DOO

Q� LI I�[�P� LV FRQYH[ LQ LWV nUVW DUJXPHQW IRU DOO P� $QI DOVR KDV WKLV SURSHUW\�� 7R VHH WKLV� QRWH

WKDW LI 9Q��c� DQG I�[�P� DUH ERWK FRQYH[ IRU DOO n[HG P� WKHQ %QI DOVR KDV WKLV SURSHUW\� 7KH

VDPH KROGV IRU &� )LQDOO\� WKH PD[LPXP RI WZR FRQYH[ IXQFWLRQV RI p LV LWVHOI D FRQYH[ IXQFWLRQ RI

p� 7KXV� LI 9Q��c� LV FRQYH[ LQ p IRU n[HG P� WKHQ $QI DOVR LQKHULWV WKLV SURSHUW\� 7KLV LPSOLHV

�7R SURYH WKDW WKLV LV VXpFLHQW IRU VKRZLQJ WKH GHVLUHG UHVXOW� QRWH WKDW IRU DQ DUELWUDU\ I � '�MM$QI b 9QMM MM$QI b$Q9QMM � �MMI b 9QMM

EHFDXVH $Q LV D FRQWUDFWLRQ DQG 9Q�c� LV LWV n[HG SRLQW� +HUH� ZH KDYH XVHG MMI b JMM WR GHQRWH WKH GLVWDQFH EHWZHHQI � & DQG J � &� )RU P w �� OHW $P

Q I s $Q�$Pb�Q I�� 7KHQ� WKH DERYH HTXDWLRQ LPSOLHV WKDW MM$P

Q I b 9QMM ��PMMI b 9QMM ZKLFK FRQYHUJHV WR ]HUR DV P � �� 7KHUHIRUH� WKH IXQFWLRQ WKDW LV WKH OLPLW RI $P

Q I HTXDOV 9Q�c� IRUDQ\ DUELWUDU\ I � '� 1RZ VXSSRVH WKDW IRU DQ\ I � ' ZKLFK LV FRQYH[ LQ [� LW LV WKH FDVH WKDW $QI DOVR KDV WKLVSURSHUW\� 7KHQ� $PI KDV WKLV SURSHUW\ IRU DOO P� 7KHUHIRUH� IRU w � >�� @�

9Q�w[� ��b w�\�P�

OLPP��

�$PQ I��w[� ��b w�\�P�

� w OLPP��

�$PQ I��[�P� � ��b w� OLP

P��$P

Q I��\�P�

w9Q�[�P� � ��b w�9Q�\�P��

��

WKDW LI 9Q��c� LV FRQYH[ LQ p IRU n[HG P� 9Q�c� ZLOO EH WRR� 7R FRPSOHWH WKH SURRI QRWH WKDW 91 �c��EHLQJ OLQHDU LQ p LV DOVR FRQYH[ LQ p�

7R SURYH WKDW 9Q�p�P� LV �ZHDNO\� LQFUHDVLQJ LQ p IRU n[HG P� OHW D ! �� :KHQ D �� WKLV

SURSHUW\ IROORZV IURP WKH OLQHDULW\ RI 9Q�c� LQ p�� ,W FDQ EH FKHFNHG E\ UHFXUVLYH VXEVWLWXWLRQ WKDW

IRU Q �� 1 b �� 9Q�p�P� p*Q�[�P� ZKHUH [ Dp DQG *Q � 5� d1� � �� LV WKH XQLTXH

n[HG SRLQW RI D PDSSLQJ +Q IURP %� WKH VHW RI ERXQGHG DQG FRQWLQXRXV IXQFWLRQV RQ = WR %�GHnQHG E\� �+QI��[�P� PD[>�5QI��[�P�� 6I��[�P�@ ZKHUH �5QI��[�P� HTXDOV

nx{�Q�P�

=*Q��[ H[S�b\��P�K�\�G\ � n>�b x{�Q�P�@

=I�[ H[S�\��P� ��K�\�G\ b [b E�

�6I��[�P� n

=I�[ H[S�b\��P�K�\�G\�

DQG ZKHUH K�\� LV WKH QRUPDO GHQVLW\ ZLWK PHDQ xb w� ��}� DQG YDULDQFH JLYHQ E\ }�� ,W LV QRZ

VHHQ WKDW LI *Q��[�P� LV GHFUHDVLQJ LQ [ DQG LI I � % LV GHFUHDVLQJ LQ LWV nUVW DUJXPHQW� WKHQ 5QI

DQG 6I �DQG FRQVHTXHQWO\ +Q� KDYH WKLV SURSHUW\ WRR� 7KLV LPSOLHV WKDW LI *Q��c� LV GHFUHDVLQJ LQ[ WKHQ VR LV *Q�c�� )LQDOO\� UHFDOO WKDW *1 �c� LV ZHDNO\ GHFUHDVLQJ WULYLDOO\ EHFDXVH LW LV FRQVWDQW�7KXV� *Q�c� LV GHFUHDVLQJ LQ [ IRU DOO Q� 7KLV LPSOLHV WKDW 9Q�p�P� s p*�[�P� p*Q�

Dp �P� LV

LQFUHDVLQJ LQ p�

7KH VHFRQG SDUW RI WKH SURSRVLWLRQ LV WKH VWDWHPHQW WKDW 9 �Q � ��P� p� �� w 9 �Q�P� p� �� w9 �Q�P�� p�� 7KH SURRI LV VLPLODU WR WKH nUVW SDUW� &RQVLGHU nUVW WKH PRQRWRQLFLW\ LQ Q IRU n[HG

P DQG p� )RU D n[HG Q� FRQVLGHU I � ' WKDW VDWLVI\ WKH FRQGLWLRQ� I�[�P� � 9Q��[�P�� ,W FDQ

WKHQ EH VKRZQ WKDW $QI DOVR KDV WKLV SURSHUW\� 7KLV LPSOLHV WKDW 9 �c�P� p� �� LV LQFUHDVLQJ LQ Q IRU

DOO P DQG p� 7KH GHWDLOV RI WKLV DUJXPHQW IROORZ�

6XSSRVH DV WKH LQGXFWLRQ K\SRWKHVLV� WKDW 91 �p�P� w 91b��[�P� w � � � w 9Q��[�P�� :H QHHG

WR VKRZ WKDW WKLV LPSOLHV 9Q��[�P� w 9Q�[�P�� 3LFN DQ DUELWUDU\ I � ' VXFK WKDW I�[�P� �9Q��[�P�� 5HFDOO WKDW �$QI��[�P� PD[>�%QI��[�P�� &I��[�P�@� 0RUHRYHU� �%QI��[�P�

HTXDOV

�x{�Q�P�

=9Q��[ H[S�\��P�Q�\�G\ � �>�b x{�Q�P�@

=I�[ H[S�\��P� ��Q�\�G\ b Db E[

� �x{�Q�P�

=9Q��[ H[S�\��P�Q�\�G\ � �>�b x{�Q�P�@

=9Q��[ H[S�\��P� ��Q�\�G\ b Db E[

�%Q��9Q��[�P�

ZKHUH ZH KDYH XVHG WKH LQGXFWLRQ K\SRWKHVLV DQG WKH IDFW WKDW I�c� � 9Q��c�� 6LPLODUO\� �&I��c� ��&9Q��c�� 7KHUHIRUH� �$QI��c� � �$Q��9Q��c� 9Q��c�� 7KLV LPSOLHV WKDW 9Q�c� � 9Q��c� LIWKH LQGXFWLRQ K\SRWKHVLV KROGV� 7R VWDUW WKH LQGXFWLRQ� LW IROORZV IURP GLUHFW FRPSXWDWLRQV WKDW

91b��c� � 91 �c��6LPLODUO\� LW FDQ EH VKRZQ WKDW LI 9Q��p�P� LV GHFUHDVLQJ LQP IRU DOO p DQG LI I � ' DOVR VDWLVnHV

WKLV SURSHUW\ DQG WKH SURSHUW\ WKDW I�[�P� � 9Q��[�P�� WKHQ %QI DQG &I �DQG FRQVHTXHQWO\

�7KH H[LVWHQFH RI D XQLTXH n[HG SRLQW IRU +Q IROORZV IURP WKH DUJXPHQWV SUHVHQWHG DERYH IRU 9Q�c��

��

$QI� PXVW DOVR EH GHFUHDVLQJ LQ P IRU DQ\ p� 7KLV LPSOLHV WKDW 9 �Q�P� p� �� LV GHFUHDVLQJ LQ P IRU

DOO Q DQG p VLQFH 9 �1�P� p� �� GRHV QRW GHSHQG RQ P�

)LQDOO\� ZH QHHG WR SURYH WKH ODVW SDUW RI WKH SURSRVLWLRQ� WKDW OLPp�� 9 �Q�P� p� �� 7KLV

IROORZV GLUHFWO\ IURP WKH IDFWV WKDW � � 9 �Q�P� p� � 9 �1�P� p� DQG WKDW 9 �1�P� p�� DV p � ��


)LUVW ZH PXVW HVWDEOLVK WKDW LI �� KROGV WKHQ �� IROORZV� 6XEVWLWXWLRQ IURP �� DQG ��

FRPELQLQJ H[SRQHQWV� DQG DSSO\LQJ WKH IRUPXOD IRU WKH PHDQ RI D ORJQRUPDO YDULDEOH JLYHV�

(W

|]�W� ��

]�W�p�W� ��[�W� ��

} H[S�xb U b w��

7KHQ VXEVWLWXWLQJ WKH FRQMHFWXUHG IRUP IRU WKH YDOXH IXQFWLRQ� �� LQWR WKH GHnQLWLRQ RI 9H� ��

DQG UHFRJQL]LQJ WKDW p�W�� LV LQGHSHQGHQW RI [�W�� JLYHV �� 7KH UHVXOW QRZ IROORZV E\ VLPSO\

VXEVWLWXWLQJ WKH FRQMHFWXUHG IRUPV IRU 9 DQG 9H LQWR �� ZLWK D � DQG UHFRJQL]LQJ WKDW ��

KROGV�


7KH SURRI RI WKH nUVW SDUW RI WKH SURSRVLWLRQ IROORZV LPPHGLDWHO\ IURP WKH IDFW WKDW WKH VHFRQG

DUJXPHQW RI WKH PD[>c� c@ RSHUDWRU LQ �� LV WKH GLVFRXQWHG YDOXH RI WKH OHIW�KDQG VLGH RI WKH HTXDWLRQ�LI WKH YDOXH LV nQLWH� n � �� 7KH IXQFWLRQ K�Q�P� FDQQRW HTXDO WKH GLVFRXQWHG YDOXH RI LWVHOI�

nK�Q�P�� XQOHVV ERWK DUH ]HUR�

7KDW K�1�P� ��bn LV D GLUHFW FRQVHTXHQFH RI 3URSRVLWLRQ � DQG 3URSRVLWLRQ ��

7KH nQLWHQHVV RI Pe�c� IROORZV IURP WKH IDFW ZKHQ E ! �� WKHUH H[LVWV D nQLWH P VXFK WKDW

WKH nUVW DUJXPHQW RI WKH PD[>c� c@ RSHUDWRU LQ WKH ULJKW�KDQG VLGH RI �� LV VWULFWO\ OHVV WKDQ ]HUR�:H HVWDEOLVK WKLV E\ LQGXFWLRQ RQ Q� 7DNH Q 1 b �� :H HVWDEOLVKHG DERYH WKDW K�1�P� GRHV

QRW GHSHQG RQ P� &OHDUO\� OLPP�� x{�1 b ��P� �� ,I WKH nUVW DUJXPHQW LQ �� QHYHU EHFDPHQHJDWLYH� WKHQ WKH PD[>c� c@ RSHUDWRU FRXOG EH GURSSHG DQG ZH ZRXOG KDYH

K�Q�P�b x{�1 b ��P� n

�b nb ��b x{�Q�P��nK�1 b ��P� �� E ��

7DNLQJ WKH OLPLW DV P JRHV WR LQnQLW\ SURYLGHV�

OLPP��K�1 b ��P�b nK�1 b ��P� �� E � ��

ZKLFK LPSOLHV WKDW IRU ODUJH HQRXJK P� K�1b��P� LV LQFUHDVLQJ LQ P� 7KLV FRQWUDGLFWV 3URSRVLWLRQ

��

� DQG VR HVWDEOLVKHV WKH nQLWHQHVV RI Pe�1b�� 1RZ FRQVLGHU Q � 1b�� 6XSSRVH DV WKH LQGXFWLRQK\SRWKHVLV WKDW Pe�Q�� LV nQLWH� 8VLQJ VLPLODU DUJXPHQWV ZH KDYH WKDW LI� IRU D JLYHQ Q� WKH nUVW

DUJXPHQW RI WKH PD[>c� c@ RSHUDWRU GLG QRW HYHQWXDOO\ EHFRPH QHJDWLYH� WKHQ

OLPP��K�Q�P�b nK�Q�P� �� E � ��

ZKLFK FRQWUDGLFWV 3URSRVLWLRQ � IRU WKH VDPH UHDVRQ�


1RWH WKDW WKH GHnQLWLRQ RI Pe LV MXVW D UHVWDWHPHQW RI �� 7KXV WKH FDVH ZKHQ Q � 1 DQG

P ! Pe�Q� LV WULYLDOO\ WUXH� 6R LV WKH FDVH ZKHQ Q 1 � :H ZLOO VKRZ WKDW WKH UHPDLQLQJ IXQFWLRQDO

IRUP VDWLVnHV HTXDWLRQ �� DERYH�

)LUVW QRWH WKDW WKH UDWLR RI 3RFKKDPPHUV LQ WKLV IXQFWLRQDO IRUP

GHULYHV IURP WKH IROORZLQJ IDFW�

rP� m�

sLr

P� Q� m� � m�sL

L<

M �

��b x{�Q�P� M b ��

8VLQJ �� DQG WKH GHnQLWLRQ RI x{ �HTXDWLRQ �� WKH IXQFWLRQDO IRUP IRU WKH ODVW FDVH FDQ EH

ZULWWHQ DV

K�Q�P� ��

Pe�Q�bPb�;L �

>nx{�Q�P� L� ��K�Q� ��P� L� ��b E@ nLL<

M �

r�b x{�Q�P� M�

s

Pe�Q�bP;L �

>nx{�Q�P� L�K�Q� ��P� L�b E@ nLb�L<

M �

r�b x{�Q�P� M b ��

s

3Pe�Q�bPL � >nx{�Q�P� L�K�Q� ��P� L�b E@ nL 4L

M �

r�b x{�Q�P� M b ��

snr�b x{�Q�P�

s

��#Pe�Q�bP;

L �

>nx{�Q�P� L�K�Q� ��P� L�b E@ nLL<

M �

r�b x{�Q�P� M b ��

s�$

brnx{�Q�P�K�Q� ��P�b E

s��

nr�b x{�Q�P�

s

K�Q�P�b nx{�Q�P�K�Q� ��P� � E

nr�b x{�Q�P�

s �

��

6ROYLQJ IRU K�Q�P� SURYLGHV

K�Q�P� nr�b x{�Q�P�

sK�Q�P� �� nx{�Q�P�K�Q� ��P�b E ��

ZKLFK LV �� ZKHQ P � Pe�Q��


,I J�� LW LV VXERSWLPDO WR VWDUW WKH 5'� 6LQFH WKH 5' LV QHYHU VWDUWHG� LWV YDOXH LV

DOZD\V ]HUR�

,W LV VWUDLJKWIRUZDUG WR YHULI\ WKDW WKH IROORZLQJ IXQFWLRQ VDWLVnHV ��

J�1 b Q� o1bQ �

�b nb Et

1bQb�;L �

oL ��

7KH H[SUHVVLRQV IRU J�Q� DQG J�� WKHQ IROORZ IURP UHZULWLQJ WKH DERYH XVLQJ WKH IRUPXOD IRU D

nQLWH JHRPHWULF VHULHV� )LQDOO\� QRWH WKDW WKH TXDQWLW\K��bn �

Et�bo

LLV D SRVLWLYH FRQVWDQW� VLQFH n�

o DQG t DUH DOO SRVLWLYH IUDFWLRQV� 6R�

J��Q� bo�1bQ��

�b n�

Et

�b o

wORJ o ! � ��

DQG

J��Q� o�1bQ��ORJ o��

�b n�

Et

�b o

w! ��


,W FDQ EH YHULnHG E\ GLUHFW VXEVWLWXWLRQ LQ WKH IXQFWLRQDO HTXDWLRQV GHnQLQJ *Q�c� LQ WKH SURRIRI 3URSRVLWLRQ � DERYH WKDW

OLPp��

9 �Q�P� p� ��

p *Q��P� K�Q�P�

ZKHUH K�Q�P� LV GHnQHG E\ WKH UHFXUVLRQ LQ �� 7KLV VD\V WKDW DV p � �� WKH HmHFW RI n[HGFRVW D EHFRPHV QHJOLJLEOH RQ WKH YDOXH RI WKH SURMHFW DQG LW FRQYHUJHV WR WKH YDOXH LQ WKH FDVH ZLWK

D ��


��

7KH VHFRQG SDUW RI WKH SURSRVLWLRQ LV MXVW D UHVWDWHPHQW RI 3URSRVLWLRQ �� )URP 3URSRVLWLRQ �

ZH NQRZ WKDW DV p � �� WKH YDOXH RI WKH SURMHFW FRQYHUJHV WR WKH YDOXH LQ WKH SURSRUWLRQDO FDVH�%XW ZLWK E �� WKLV YDOXH ZLOO EH SRVLWLYH� 7KLV ZLOO� EHFDXVH p�W� LV D JHRPHWULF UDQGRP ZDON�

KDSSHQ ZLWK SRVLWLYH SUREDELOLW\� 6LQFH WKHUH LV QR FRVW DVVRFLDWHG ZLWK ZDLWLQJ ZKLOH WKH SURMHFW

LV PRWKEDOOHG� LW FDQQRW EH RSWLPDO WR DEDQGRQ WKH SURMHFW�

3URRI RI 3URSRVLWLRQ ��

:H nUVW HVWDEOLVK WKDW IRU Pe�Q� GHnQHG LQ 3URSRVLWLRQ � ZKHQ P ! Pe�Q� WKH SURMHFW LVDEDQGRQHG� WKDW LV� 9 �P�Q� p� �� :KHQ D �� ZH NQRZ WKDW 9 �P�Q� p� pK�Q�P�� )URP

3URSRVLWLRQ � ZH NQRZ WKHUH LV DOZD\V DQ P VXpFLHQWO\ ODUJH WKDW WKH SURMHFW LV DEDQGRQHG� VR

OLPP�� K�Q�P� � IRU Q � 1 b �� 7KH UHVXOW WKHQ IROORZV IURP WKH IDFW WKDW ZKHQ D ! ��

� � 9 �Q�P� p� � K�P�Q�p�

1H[W ZH VKRZ WKDW LI P � Pe�Q�� WKH SURMHFW LV QHYHU DEDQGRQHG� )LUVW QRWH WKDW IRU WKH QRn[HG FRVW FDVH� LI P � Pe�Q� WKHQ 9Q�Q�P� p� ! �� ZKHUH 9Q GHQRWHV WKH YDOXH ZKHQ D �� 8VLQJ

WKLV UHVXOW DQG 3URSRVLWLRQ �� ZH KDYH�

OLPp��

9 �Q�P� p� 9Q�Q�P� p� ! � ��

7KXV� WKHUH PXVW DOZD\V H[LVW D p ODUJH HQRXJK VR WKDW IRU DQ\ P � Pe�Q��9 �Q�P� p� ! �� ,W FDQ

QHYHU EH RSWLPDO WR DEDQGRQ D SURMHFW IRU ZKLFK DQ DWWDLQDEOH VWDWH H[LVWV LQ ZKLFK WKH YDOXH RI

WKH SURMHFW LV SRVLWLYH�

:H QH[W HVWDEOLVK VRPH SURSHUWLHV RI WKH GHULYDWLYH RI 9Q�c� ZLWK UHVSHFW WR p� 6LQFH 9Q�c� LVFRQYH[ LQ p� LW LV GLmHUHQWLDEOH DOPRVW HYHU\ZKHUH� 5HFDOO IURP WKH GHnQLWLRQ RI 9Q�c� WKDW LW LV WKHPD[LPXP RI WZR IXQFWLRQV� %Q9Q DQG &9Q� ZKLFK DUH FRQYH[ DQG �FRQWLQXRXVO\� GLmHUHQWLDEOH LQ

p� 7KXV� H[FHSW ZKHQ WKHVH WZR IXQFWLRQV DUH HTXDO� 9Q�c� LV DOVR GLmHUHQWLDEOH LQ p� ,W PD\ QRW

EH GLmHUHQWLDEOH ZKHQ %Q9Q�c� DQG &9Q�c� DUH HTXDO� ,Q WKH GHULYDWLRQV EHORZ� SULPH GHQRWHV WKHGHULYDWLYH RI D IXQFWLRQ ZLWK UHVSHFW WR p LI LW LV GLmHUHQWLDEOH RU LWV ULJKW�GHULYDWLYH LI LW LV QRW� 7KH

ULJKW �DQG OHIW��GHULYDWLYHV RI 9Q�c� ZLWK UHVSHFW WR p H[LVW VLQFH 9Q�c� LV FRQYH[�1RZ ZH GHULYH DQ H[SUHVVLRQ IRU 9 �

Q�c�� 'HnQH

$R�P�Q� s Ip � ��M�%Q9Q��p�P� ! �&9Q��p�P�J� ��

OHW

$RR�P�Q� s Ip � ��M�%Q9Q��p�P� � �&9Q��p�P�J ��

DQG� nQDOO\�a$ s Ip � ��M�%Q9Q��p�P� �&9Q��p�P�J� ��

��

7KHQ� IRU p � $R�P�Q��

9 �Q�p�P� �x{�Q�P�

=9Q��H

]�P�Jp�]� p�G] � ��b {�Q�P��

=9Q�H

]�P� ��Jp�]� p�G] b E ��

nx{�Q�P�

=9 �Q��p H[S�\��P�K�\�G\ � n��b {�Q�P��

=9 �Q�p H[S�\��P� ��K�\�G\ b E

ZKHUH J�]� p� GHQRWHV WKH QRUPDO GHQVLW\ ZLWK PHDQ �x b w b ��}�� OQ�p� DQG YDULDQFH }�� Jp�c�GHQRWHV LWV GHULYDWLYH ZLWK UHVSHFW WR p� DQG ZKHUH K�c� KDV DOUHDG\ EHHQ GHnQHG DV WKH QRUPDOGHQVLW\ ZLWK PHDQ �xb w� ��}�� DQG YDULDQFH }��

)RU p � $RR�P�Q��

9 �Q�p�P� �

=9Q�H

]�P�Jp�]� p�G] n

=9 �Q�p H[S�\��P�K�\�G\ ��

)RU p � a$� 9 �Q�p�P� HTXDOV

PD[

�nx{�Q�P�

=9 �Q��p H[S�\��P�K�\�G\ � n>�b {�Q�P�@

=9 �Q�p H[S�\��P� ��K�\�G\ b E�

��

n

=9 �Q�p H[S�\��P�Q�\�G\

w��

7KH DERYH H[SUHVVLRQV IRU 9 �Q�c� FDQ EH GHGXFHG IURP WKH H[SUHVVLRQ IRU 9Q�c�� 7KH LQWHUFKDQJLQJ

RI LQWHJUDWLRQ DQG GLmHUHQWLDWLRQ WKDW WKH DERYH H[SUHVVLRQV LQYROYH FDQ EH MXVWLnHG IRUPDOO\� :H

RPLW WKH GHWDLOV� 6LPLODU H[SUHVVLRQV REWDLQ IRU WKH GHULYDWLYHV RI WKH nQLWH�KRUL]RQ DSSUR[LPDWLRQV

WR 9Q�c��7KH QH[W WZR OHPPDV SURYH UHVXOWV RQ WKH VORSH RI WKH FRPSRQHQWV RI WKH YDOXH IXQFWLRQ�

/HPPD � 9 �Q�p�P� � $

3URRI� :H FRQVLGHU WKH FDVH ZKHQ D ! �� 7KH UHVXOW IROORZV IURP GLUHFW FRPSXWDWLRQV ZKHQ

D �� 6XSSRVH p � $R�Q�P�� OHW [ Dp � 7KHQ� WKH nUVW H[SUHVVLRQ IRU 9

�Q�c� JLYHQ LQ �� DERYH�

DQG H[SOLFLW GLmHUHQWLDWLRQ RI J�]� p� ZLWK UHVSHFW WR p� LPSO\ WKDW

9 �Q�p�P�

�{�Q�P�

}�p

=9Q��H

]�P�>] b OQ p b Ax@J�]� p�G] ��>�b {�Q�P�@

}�p

=9Q�H

]�P� ��>] b OQ p b Ax@J�]� p�G] b E

{�Q�P�n

}�

=*Q��[ H[S�b\��P�>\ b Ax@K�\�G\ �

>�b {�Q�P�@n

}�

=*Q�[ H[S�b\��P� ��>\ b Ax@K�\�G\ b E

ZKHUH Ax xb wb ��}� DQG ZKHUH *Q�c�� Q �� 1� KDYH EHHQ GHnQHG LQ WKH SURRI RI 3URSRVL�

��

WLRQ �� 1RWH WKDW *Q�p�P� � $� 7KHUHIRUH� WKH ODVW H[SUHVVLRQ LQ WKH DERYH VHW RI HTXDWLRQV LV QR

JUHDWHU WKDQ

{�Q�P�n$

}�

=>\ b Ax@K�\�G\ � >�b {�Q�P�@n$

}�

=>\ b Ax@K�\�G\ n$ � $ ��

EHFDXVH K�c� LV WKH QRUPDO GHQVLW\ ZLWK PHDQ JLYHQ E\ Ax� }��

7KH VDPH FRQFOXVLRQ FDQ EH HVWDEOLVKHG IRU p � $RR�Q�P�� 7RJHWKHU WKHVH WZR FRQFOXVLRQ LPSO\

WKDW WKH VDPH KROGV IRU p � a$�Q�P��

/HPPD � �%Q9Q��p�P� w �&9Q�

��p�P�

3URRI� 7KLV UHVXOW LV SURYHG E\ nUVW SURYLQJ DQ DQDORJRXV UHVXOW IRU -Q�p�P� �� 7 �� WKH nQLWH�

KRUL]RQ DSSUR[LPDWLRQV IRU 9Q�p�P�� 5HFDOO WKDW -Q�p�P� �� 7 � PD[>.Q�p�P� �� 7 �� /Q�p�P� �� 7 �@

ZKHUH .Q�c� DQG /Q�c� DUH GHnQHG LQ WKH GHULYDWLRQ RI YDOXH IXQFWLRQ DERYH� 1RWH IRU IXWXUH

UHIHUHQFH WKDW VLQFH WKH YDOXDWLRQ SUREOHP FRQVLGHUHG KHUH LV WLPH�KRPRJHQHRXV� .Q�p�P� �� 7 � s.Q�p�P� �� 7 � �� 6LPLODUO\� IRU /Q�c�� :H VKRZ WKDW . �

Q�p�P� �� 7 � w /�Q�p�P� �� 7 � IRU DOO 7 w ��

7KH SURRI LV E\ LQGXFWLRQ RQ 7 � 6XSSRVH WKDW . �Q�c� �� 7 �� w /�

Q�c� �� 7 �� IRU VRPH 7 w �� 7KLVPHDQV WKDW

. �Q�p�P� �� 7 � ��

n{�Q�P�

=- �Q��p H[S�\��P� �� 7 �K�\�G\

�n>�b {�Q�P�@

=- �Q�p H[S�\��P� �� 7 �K�\�G\ b E

w n

=- �Q�p H[S�\��P� �� 7 �K�\�G\

/�Q�p�P� �� 7 � ��

IRU �p�P� � =� ,W FDQ EH VKRZQ WKDW WKLV LPSOLHV WKDW

. �Q�p�P� �� 7 �� w - �

Q�p�P� �� 7 � �� w /�Q�p�P� �� 7 �� n

=- �Q�p H[S�\��P� �� 7 �K�\�G\ ��

:H VKRZ WKDW WKLV LQ WXUQ LPSOLHV WKDW . �Q�c� �� 7 � �� w /�

Q�c� �� 7 � �� 8VLQJ WKH GHnQLWLRQ RI. �

Q�p�P� �� 7 � ��

. �Q�p�P� �� 7 � ��

n{�Q�P�

=- �Q��p H[S�\��P� �� 7 � ��K�\�G\

�n>�b {�Q�P�@

=- �Q�p H[S�\��P� �� 7 � ��K�\�G\ b E

��

w n�{�Q�P�

= =- �Q��p H[S�\ � ]��P� �� 7 �K�]�K�\�G]G\

� n�>�b {�Q�P�@

= =- �Q�p H[S�\ � ]��P� �� 7 �K�]�K�\�G]G\ b E

n

=. �

Q�p H[S�\��P� �� 7 � ��K�\�G\

w n

=- �Q�p H[S�\��P� �� 7 � ��G\

/�Q�p�P� �� 7 � ��

ZKHUH WKH LQHTXDOLWLHV PDNH XVH RI �� DQG WKH HTXDOLWLHV PDNH XVH RI GHnQLWLRQV� 7R FRQFOXGH WKH

LQGXFWLRQ� LW FDQ EH YHULnHG E\ GLUHFW FRPSXWDWLRQV WKDW WKH LQGXFWLRQ K\SRWKHVLV KROGV IRU 7 ��

5HFDOO QRZ WKDW OLP7��.Q�p�P� �� 7 � �%Q9Q��p�P� IRU Q � 1b� DQG OLP7�� /Q�p�P� �� 7 �

&9Q�p�P�� 0RUHRYHU� .Q�c� 7 �� /Q�c� 7 �� %Q9Q�c� DQG &9Q�c� DUH DOO GLmHUHQWLDEOH DQG FRQYH[� $VD UHVXOW� WKH GHULYDWLYHV RI /Q�c� 7 � FRQYHUJHV WR WKH GHULYDWLYH RI %Q9Q�c� DQG WKH GHULYDWLYH RI.Q�c� 7 � FRQYHUJHV WR WKH GHULYDWLYH RI &9Q�c� E\ 7KHRUHP �� RI 5RFNDIHOODU �� 7KHUHIRUH�

LW PXVW EH WKH FDVH WKDW �%Q9Q��p�P� w �&9Q�

��p�P� IRU Q � 1 b ��

8VLQJ WKHVH WZR OHPPDV ZH FDQ QRZ SURYH WKH UHVXOWV� )LUVW� ZH VKRZ LW LV RSWLPDO WR LQYHVW

ZKHQHYHU p�W� LV JUHDWHU WKDQ VRPH pe�Q�P��6XSSRVH D ! �� &RQVLGHU �P�Q� VXFK WKDW WKH YDOXH RI WKH 5' SURMHFW LV VWULFWO\ SRVLWLYH

ZKHQ D �� 7KDW LV� FRQVLGHU �Q�P�� VXFK WKDW K�Q�P� GHnQHG LQ �� LV VWULFWO\ SRVLWLYH�� 1RWH

WKDW �L� IRU DQ\ Q � 1 b �� %Q9Q��P� � � DQG &9Q��P� � DQG �LL� IRU Q � 1 b ��

OLPp��

%Q9Q�p�P�

p K�P�Q� ! OLP

p��&9Q��p�P�

p nK�Q�P��

7KDW LV� IRU n[HG �P�Q�� WKH IXQFWLRQ %Q9Q�p�P� VWDUWV EHORZ �&9Q��p�P� DQG HQGV XS VWULFWO\

KLJKHU� 0RUHRYHU� /HPPD � VKRZV WKDW WKH GHULYDWLYH RI �%Q9Q��c� �ZLWK UHVSHFW WR p� LV DOZD\V

JUHDWHU WKDQ RU HTXDO WR WKDW RI �&9Q��c�� 7KDW PHDQV WKDW WKH IXQFWLRQV� %Q9Q�p�P� DQG &9Q�p�P�

IRU n[HG �P�Q� RQO\ FURVV RQFH� 'HQRWLQJ WKH SRLQW RI LQWHUVHFWLRQ RI WKHVH WZR IXQFWLRQV DV pe�Q�P��ZH VHH WKDW WKH RSWLPDO 5' SROLF\ IRU D JLYHQ �p� Q�P� LV WR FRQWLQXH ZLWK WKH 5' LI p w pe�Q�P�DQG WR PRWKEDOO RWKHUZLVH� ,I p LV VR ORZ WKDW 9Q�p�P� �� WKHQ WKH SROLF\ LV WR PRWKEDOO WKH

SURMHFW IRUHYHU� WKDW LV WR DEDQGRQ WKH SURMHFW�

,W UHPDLQV WR VKRZ WKDW pe�Q�P� LQFUHDVHV DV WKH n[HG FRVW LQFUHDVHV�)RU D JLYHQ D ! �� DQG IRU n[HG �Q�P�� pe�Q�P�D� LV WKH XQLTXH VROXWLRQ WR WKH HTXDWLRQ�

�%Q9Q��pe�P� D� �&9Q��p

e�P� D�� 7KLV HTXDWLRQ FDQ DOVR ZULWWHQ DV pe�P�Q� D[e�P�Q� ZKHUH

[e�P�Q� LV WKH XQLTXH VROXWLRQ WR WKH HTXDWLRQ� �5Q*Q��[e�P� �6*Q��[

e�P�� 1HLWKHU 5Q*Q

QRU 6*Q GHSHQGV RQ D� 7KHUHIRUH� [e�P�Q� GRHV QRW GHSHQG RQ D� 7KHUHIRUH� ZH KDYH WKDW

pe�Q�P� D� � pe�Q�P� D�� LI D � D�� 7KDW LV� LI LW LV RSWLPDO WR PRWK�EDOO �RU DEDQGRQ� WKH SURMHFWDW D JLYHQ �p� Q�P� DQG D JLYHQ D� WKHQ WKLV UHPDLQV WKH FDVH IRU DOO KLJKHU D�

��


7KH YDOXH RI WKH nUP JLYHQ IRU WKLV FDVH E\ 3URSRVLWLRQ � LV SURSRUWLRQDO WR p�W�� 7KLV LV WKH

?FXP�FDVK�oRZ� YDOXH� /HW &�W� �� GHQRWH WKH FDVK oRZ QH[W SHULRG� ZKLFK ZLOO EH HLWKHU ]HUR RU

b,�W� �� %RWK RI WKHVH DUH SURSRUWLRQDO WR p�W� �� $FFRUGLQJO\�

[�W� ��9 >Q�W� ��P�W� �� p�W� ��@ [�W� ��p�W� ��K>Q�W� ��P�W� ��@ ��

6LQFH� JLYHQ WLPH�W LQIRUPDWLRQ� WKH LQQRYDWLRQV LQ [�W � �� DQG p�W � �� DUH LQGHSHQGHQW RI WKRVH

LQ Q�W� �� DQG P�W� �� WKH FXUUHQW H[SHFWDWLRQ RI WKLV YDOXH QH[W SHULRG LV

(WI[�W� ��p�W� ��K>Q�W� ��P�W� ��@J {\p�W� H[S�x�(WIK>Q�W� ��P�W� ��@J ��

7KH FXUUHQW �H[�FDVK�oRZ� YDOXH RI WKH nUP ZLOO WKHQ EH

(W

|]�W� ��

]�W�[�W� ��p�W� ��K>Q�W� ��P�W� ��@

} {\p�W� H[S�xbUbw�(W IK>Q�W� ��P�W� ��@J

��

EHFDXVH DOO WKH YDULDEOHV H[FHSW p�W� �� KDYH LQQRYDWLRQV WKDW� FRQGLWLRQDO RQ WLPH�W LQIRUPDWLRQ�

DUH LQGHSHQGHQW RI ]�W�� 'LYLGLQJ WKH H[SHFWHG SD\Rm� �� E\ LWV YDOXH� �� JLYHV HU�w DV WKH

SHU�GROODU H[SHFWHG UHWXUQ�

9DOXH RI WKH 3URMHFW ZKHQ ,QYHVWPHQW LV 'HWHUPLQLVWLF

/HPPD � :KHQ { LV NQRZQ DQG WKH nUP LV KDV QR RSWLRQ WR VXVSHQG IXWXUH LQYHVWPHQW� WKH YDOXH

RI WKH SURMHFW LV�

9 �Q� p� m�Q�p b o�Q� ��

ZKHUH

m�Q� �1bQ

��b n�b E

�b ��b {�n

��b �1bQ

�b �

��

o�Q�

tD

�b ��b {��

u��b �1bQ

�b �

��

DQG ZKHUH � s {\HbU� � s {�

�b��b{�� DQG � s {n�b��b{�n �

3URRI� :LWK QR RSWLRQ FRPSRQHQW WKH YDOXH IXQFWLRQ MXVW VROYHV�

9 �Q� p� {9H�Q� �� p� � ��b {�9H�Q� p�b Ep b D ��

��

7KH FRQMHFWXUH WKDW WKH VROXWLRQ KDV WKH IRUP GHVFULEHG LQ WKH SURSRVLWLRQ LV YHULnHG VLPSO\ WKURXJK

UHFXUVLYH VXEVWLWXWLRQ DQG VLPSOLnFDWLRQ�


:H SURYH WKLV UHVXOW IRU WKH FDVH ZKHUH WKH SUREDELOLW\ RI WHFKQLFDO VXFFHVV DW HDFK VWDJH LV

NQRZQ� 6XSSRVH WKH SUREDELOLWLHV RI VXFFHVV DQG RI REVROHVFHQFH RYHU DQ\ LQWHUYDO RI WLPH� c� DUH

WDNHQ WR EH SURSRUWLRQDO WR c� ,QYHVWPHQW RFFXUV DW UDWH ,� VR WKH LQYHVWPHQW RI WLPH LQWHUYDO c

LV ,c� 7KH FDVK oRZ DW W�c� p�W�c�� LV JLYHQ E\

p�W� H[S>�xb �

�}��c � }

Scq�W�c�@ ��

ZKHUH q�W� LV L�L�G� QRUPDO �� 7KH FRQWLQXRXVO\ FRPSRXQGHG ULVNOHVV UDWH LV U� $V c JRHV

WR �� WKH SURFHVV IRU p�W� FRQYHUJHV WR D JHRPHWULF %URZQLDQ PRWLRQ� 2XU GLVFUHWH�WLPH PRGHO

FRUUHVSRQGV WR c �� 7KH SUREDELOLW\ RI VXFFHVV RYHU DQ LQWHUYDO RI OHQJWK c LV WDNHQ WR EH

{c DQG WKH SUREDELOLW\ RI REVROHVFHQFH LV D\c �VR {\ LQ WKH GLVFUHWH�WLPH PRGHO FRUUHVSRQGV WR

��b D\c��

8QGHU WKH ULVN QHXWUDO SUREDELOLW\ PHDVXUH �4� LPSOLHG E\ WKH SULFLQJ NHUQHO ZH KDYH DVVXPHG�

WKH ULVN�QHXWUDO GULIW RI p�W� LQ WKH OLPLWLQJ FDVH ZRXOG EH Ax s x b w DQG LWV YRODWLOLW\ SDUDPHWHU

ZRXOG EH XQFKDQJHG�

7KH GLVFUHWH�WLPH YDOXDWLRQ HTXDWLRQ IRU VWDJH Q LV ZULWWHQ DV

9 �p� Q� PD[>��b D\c�$�� b D\c�$�@ ��

ZKHUH

$� {cH[S�bUc�(4W 9 �Q� �� p�W�c�� b {c� H[S�bUc�(4

W 9 >Q� p�W�c�@b ,c ��

DQG

$� H[S�bUc�(4W 9 >Q� p�W�c�@� ��

,Q WKH FRQWLQXRXV WLPH OLPLW� (4W 9 >Q� �� p�W�c�@ LV HTXDO WR

9 �Q� �� p�W�� 9��Q� �� p�W��Axp�W�c ��

�9��Q� �� p�W��}

�pp��W�c � R�c��

$ VLPLODU H[SUHVVLRQ GHVFULEHV 9 �Q� c�� 6XSSRVH WKDW 9 �Q � �� c� KDV DOUHDG\ EHHQ REWDLQHG� 7KHIROORZLQJ GHULYHV D GLmHUHQWLDO HTXDWLRQ IRU 9 �Q� c�� ,Q WKH UHJLRQ ZKHUH UHVHDUFK LV FRQWLQXHG� ZHKDYH

9 �p� Q� ��

��

{9 �Q� �� p�c � >�b {c@ H[S�bUc�>�b D\c@>9 �Q� p� � 9��Q� p�Axpc��

�9��Q� p�}

�pp�c@b ,c� R�c��

0RUHRYHU� LQ WKH PRWKEDOO UHJLRQ� ZH KDYH

9 �p� Q� ��

H[S>bUc@>�b D\c@>9 �Q� p� � 9��Q� p�Axpc��

�9��Q� p�}

�pp�c@ � R�c��

'LYLGLQJ ERWK VLGHV E\ c DQG WDNLQJ OLPLWV� ZH nQG WKDW LQ WKH FRQWLQXDWLRQ UHJLRQ� 9 �Q� c� VDWLVnHVWKH IROORZLQJ GLmHUHQWLDO HTXDWLRQ�

9��Q� c�Axp � ��9��Q� c�}�pp� b �U � D\ � {�9 �Q� c� � {9 �Q� �� c�b , ��

DQG LQ WKH PRWKEDOO UHJLRQ� 9 �Q� c� VDWLVnHV

9��Q� c�Axp � ��9��Q� c�}�pp� b �U � D\�9 �Q� c� ��

:H NQRZ WKDW WKH FRQWLQXDWLRQ UHJLRQ LV RI WKH IRUP p w pe�Q�� 7KXV� VROYLQJ IRU 9 �Q� c� LQYROYHVVROYLQJ IRU pe�Q� DQG WKH DERYH WZR GLmHUHQWLDO HTXDWLRQV� RQH KROGLQJ IRU p ! pe�Q� DQG WKH RWKHUIRU p � pe�Q�� 7R GHWHUPLQH pe�Q� ZH ZRXOG KDYH WR XVH YDOXH�PDWFKLQJ �FRQWLQXLW\ RI 9 �Q� c�� DQGVPRRWK SDVWLQJ �GLmHUHQWLDELOLW\ RI 9 �Q� c�� FRQGLWLRQV� )RU WKH UHVXOW EHORZ� LW LV QRW QHFHVVDU\ WRNQRZ ZKDW pe�Q� LV� EHFDXVH RXU OLPLWLQJ UHVXOW RQ ULVN SUHPLXP LV WUXH IRU VPDOO p� L�H� ZKHQ ZH

DUH LQ WKH PRWKEDOO UHJLRQ� ,Q WKLV UHJLRQ� WKH VROXWLRQ WR WKH GLmHUHQWLDO HTXDWLRQ IRU 9 �Q� c� LV RIWKH IRUP $�Q�p� ZKHUH � bE�

SE��DF�D ZLWK D ��}�p � E Ax b ��}�p DQG F �U � D\�� 7KLV� LQ

WXUQ� LPSOLHV WKDW LQ WKH HQWLUH PRWKEDOO UHJLRQ� WKH ULVN SUHPLXP RQ WKH 5' SURMHFW LV HTXDO WR

�w �VLQFH �wGW bFRYW�G9 �Q�c�9 �Q�c� �

G]] � ZKHUH ]�W� LV WKH SULFLQJ NHUQHO��


/HW 5W��Q�P� EH WKH FRQWLQXRXVO\ FRPSRXQGHG UHWXUQ IURP W WR W � � RI D SURMHFW ZLWK Q

FRPSOHWHG VWDJHV DQG P IDLOXUHV� /HW WKH QXPEHU RI FRPSOHWHG VWDJHV �IDLOXUHV� DW WLPH W � � EH

GHQRWHG Q�W� �� P�W� �� 7KHQ� FRQGLWLRQDO RQ VXUYLYDO �[�W� ��

5W��Q�P� ORJ

tK�Q�W� ��P�W� ��p�W� ��

K�Q�P�p�W�

u

ORJ

�K�Q�W� ��P�W� ��p�W�

K�Q�P�p�W�H[S

�xb }�p

�� }pqp�W� ��

��

.�Q�P� � ORJ>K�Q�W� ��P�W� ��@ � }pqp�W� ��

7KH UHVXOW IROORZV EHFDXVH WKH nUVW WHUP LQ WKH DERYH H[SUHVVLRQ LV D FRQVWDQW DQG WKH ODVW LV D

UDQGRP YDULDEOH ZLWK GLVWULEXWLRQ 1>�� }p@� 7KH PLGGOH WHUP LV D GLVFUHWH UDQGRP YDULDEOH WKDW

��

WDNHV RQ WKH YDOXH ORJ>K�Q � ��P�@ ZLWK SUREDELOLW\ x{�Q�P� DQG WKH YDOXH ORJ>K�Q�P � ��@ ZLWK

SUREDELOLW\ �b x{�Q�P��

% 1XPHULFDO 'HWDLOV

2XU SURJUDPV VROYH QXPHULFDOO\ IRU WKH YDOXH RI WKH 5' YHQWXUH DW HDFK Q� P DQG RYHU D UDQJH RI

GLVFUHWH YDOXHV IRU p� 7KH VROXWLRQ WHFKQLTXH ZH XVH LV WR UHFXUVLYHO\ VROYH �� EDFNZDUGV VWDUWLQJ

ZLWK WKH YDOXH RI WKH FRPSOHWHG SURMHFW DQG WKH YDOXH RI DQ DEDQGRQHG SURMHFW� ERWK RI ZKLFK

DUH NQRZQ DQDO\WLFDOO\� 6SHFLnFDOO\� 9 �1� c� p� LV JLYHQ E\ 3URSRVLWLRQ �� 3URSRVLWLRQ � LV XVHG WRGHULYH Pe�Q� IRU DOO YDOXHV RI Q� 8VLQJ 3URSRVLWLRQ �� ZH KDYH WKDW IRU DOO P ! Pe�Q� DQG Q � 1 �

9 �Q�P� p� �� 9 �1 b ��Pe�1 b �� p� FDQ WKHQ EH LPSOLFLWO\ GHnQHG XVLQJ ��

9 �1 b ��Pe�1 b �� p� PD[ Ix{�Q�P�9H�1� c� p�� 9H�1 b ��Pe�1 b �� p�J �

7KLV HTXDWLRQ LV WKHQ VROYHG QXPHULFDOO\ E\ nUVW LQLWLDOO\ JXHVVLQJ WKH YDOXH RI 9 �1b��Pe�1b�� p�RYHU D GLVFUHWH VHW RI YDOXHV �JULG SRLQWV� IRU p� 9DOXHV EHWZHHQ WKHVH JULG SRLQWV DUH GHULYHG E\

TXDGUDWLF LQWHUSRODWLRQ� 8VLQJ WKLV JXHVV D QHZ YDOXH RI 9 �1 b ��Pe�1 b �� p� DW HDFK JULG SRLQWLV GHULYHG DQG WKLV YDOXH LV XVHG DV WKH QHZ LQSXW� 7KLV SURFHVV LV UHSHDWHG XQWLO WKH YDOXH RYHU WKH

JULG SRLQWV FRQYHUJH �WKH QHZ YDOXH LV QR PRUH WKDQ �� GLmHUHQW IURP WKH ROG YDOXH DQ\ZKHUH

LQ WKH JULG�� 7KLV SURFHVV LV WKHQ UHSHDWHG IRU DOO UHPDLQLQJ YDOXHV RI P � Pe�1 b �� 2QFH WKLV LVFRPSOHWH� RQH PRUH VWHS EDFN FDQ EH WDNHQ DQG WKH YDOXH IRU 9 �1b��Pe�1b�� p� FDQ EH GHULYHGXVLQJ WKH VDPH SURFHVV� 7KLV LV FRQWLQXHG XQWLO 9 LV GHULYHG IRU DOO Q DQG P DW HYHU\ JULG SRLQW�

7KH RQO\ UHPDLQLQJ LVVXH LV ZKDW LV GRQH ZKHQ WKH YDOXH LV QHHGHG IRU D p RXWVLGH RI WKH JULG� 6LQFH

RXU JULGV VWDUW DW �� WKLV RQO\ RFFXUV IRU YDOXHV RI p ODUJHU WKDQ WKH PD[LPXP JULG YDOXH� 6LQFH IRU

ODUJH p� WKH PRWKEDOO RSWLRQ YDOXH LV VPDOO� VR ZH XVH WKH YDOXH RI WKH SURMHFW ZLWK QR RSWLRQ ZKLFK

FDQ EH VROYHG IRU DQDO\WLFDOO\� 7KH IROORZLQJ SURSRVLWLRQ� ZKLFK LV VWDWHG ZLWKRXW SURRI� JLYHV WKLV

VROXWLRQ�

3URSRVLWLRQ �� ,I WKHUH LV QR RSWLRQ WR KDOW GHYHORSPHQW �VR WKDW XQOHVV WKH SURMHFW LV H[WLQJXLVKHG

WKURXJK REVROHVFHQFH� WKH LQYHVWPHQW LV UHTXLUHG WR EH PDGH HDFK SHULRG� WKHQ WKH YDOXH IXQFWLRQ LV

9 �Q�P� p� AK�Q�P�p � AJ�Q�P�

VR

9H�Q�P� p� AK�Q�P�np � �AJ�Q�P�

ZKHUH AK�Q�P� DQG AJ�Q�P�� VWDWHG UHFXUVLYHO\� DUH

AK�Q�P� x{�Q�P�nAK�Q� ��P� � ��b x{�Q�P��nAK�Q�P� ��b E

AJ�Q�P� x{�Q�P��AJ�Q� ��P� � ��b x{�Q�P��AJ�Q�P� ��b D

��

ZLWK ERXQGDU\ FRQGLWLRQV AK�Q�Pe�Q� � �� AJ�Q�Pe�Q� � �� AJ�1� c� � DQG AK�1� c� J�1��

,Q RXU VLPXODWLRQV ZH XVH D �� SRLQW JULG EHWZHHQ WKH SRLQWV � DQG �� 7KH VSDFLQJ EHWZHHQ

WKH SRLQWV ZDV QRW HTXDO� PRUH SRLQWV ZHUH XVHG DW ORZHU YDOXHV RI p� 7KH UDWLR EHWZHHQ WKH ODUJHVW

VSDFLQJ EHWZHHQ JULG SRLQWV DQG WKH VPDOOHVW ZDV ��

��

5HIHUHQFHV

%HUN� -�� 5� &� *UHHQ DQG 9� 1DLN �� ?2SWLPDO ,QYHVWPHQW� *URZWK 2SWLRQV DQG 6HFXULW\

5HWXUQV�� :RUNLQJ 3DSHU� 6FKRRO RI %XVLQHVV $GPLQLVWUDWLRQ� 8QLYHUVLW\ RI :DVKLQJWRQ�

%KDWWDFKDU\D� 6� DQG '� 0RRNKHUMHH �� ?3RUWIROLR &KRLFH LQ 5HVHDUFK DQG 'HYHORSPHQW��

5DQG -RXUQDO RI (FRQRPLFV� ��

%UHDOH\� 5� $� DQG 6� &� 0\HUV �� 3ULQFLSOHV RI &RUSRUDWH )LQDQFH� 0F*UDZ�+LOO� ,QF�

%UHDOH\� 5� $�� 6� &� 0\HUV DQG $� -� 0DUFXV �� )XQGDPHQWDOV RI &RUSRUDWH )LQDQFH� 0F*UDZ�

+LOO� ,QF�

&KLOGV� 3� '� DQG $� -� 7ULDQWLV �� ?'\QDPLF 5' ,QYHVWPHQW 3ROLFLHV�� :RUNLQJ 3DSHU�

&ROOHJH RI %XVLQHVV DQG 0DQDJHPHQW� 8QLYHUVLW\ RI 0DU\ODQG�

'DVJXSWD� 3� DQG (� 0DVNLQ �� ?7KH 6LPSOH (FRQRPLFV RI 5HVHDUFK 3RUWIROLRV�� 7KH (FRQRPLF

-RXUQDO� ��

0� +� 'H*URRW �� 2SWLPDO 6WDWLVWLFDO 'HFLVLRQV� 0F*UDZ�+LOO� ,QF�

'XWWD� 3� .� �� ?2SWLPDO PDQDJHPHQW RI DQ 5' EXGJHW�� -RXUQDO RI (FRQRPLF '\QDPLFV

DQG &RQWURO� ��

*URVVPDQ� *� 0� DQG &� 6KDSLUR �� ?2SWLPDO G\QDPLF 5' SURJUDPV�� 5DQG -RXUQDO RI

(FRQRPLFV� ��

*UXYHU� *� :� �� ?2SWLPDO 5' SROLF\ IRU D SDWHQW UDFH ZLWK XQFHUWDLQ GXUDWLRQ�� 0DWKH�

PDWLFDO 6RFLDO 6FLHQFHV� ��

0DMG� 6� DQG 5� 6� 3LQG\FN �� ?7LPH WR %XLOG� 2SWLRQ 9DOXH� DQG ,QYHVWPHQW 'HFLVLRQV��

-RXUQDO RI )LQDQFLDO (FRQRPLFV� ��

0\HUV� 6� &� DQG &� '� +RZH �� ?$ /LIH�&\FOH 0RGHO RI 3KDUPDFHXWLFDO 5'�� 7HFKQLFDO

5HSRUW� 3URJUDP RQ WKH 3KDUPDFHXWLFDO ,QGXVWU\� 0DVVDFKXVHWWV ,QVWLWXWH RI 7HFKQRORJ\�

3DNHV� $�� ?3DWHQWV DV 2SWLRQV� 6RPH (VWLPDWHV RI WKH 9DOXH RI +ROGLQJ (XURSHDQ 3DWHQW 6WRFNV��

(FRQRPHWULFD� ��

3LQG\FN� 5� 6� �� ?,QYHVWPHQWV RI 8QFHUWDLQ &RVW�� -RXUQDO RI )LQDQFLDO (FRQRPLFV� ��

3RVQHU� 0� -� 0� DQG '� =XFNHUPDQ� ?2SWLPDO 5' 3URJUDPV LQ D 5DQGRP (QYLURQPHQW�� -RXUQDO

RI $SSOLHG 3UREDELOLW\� ��

5RFNDIHOODU� 5� 7� �� &RQYH[ $QDO\VLV� 3ULQFHWRQ 8QLYHUVLW\ 3UHVV�

��

6FKZDUW]� (� 6� DQG 0� 0RRQ �� ?(YDOXDWLQJ 5HVHDUFK DQG 'HYHORSPHQW LQYHVWPHQWV�� :RUN�

LQJ 3DSHU� $QGHUVRQ *UDGXDWH 6FKRRO RI 0DQDJHPHQW DW 8&/$�

:HLW]PDQ� 0� /� �� ?2SWLPDO 6HDUFK IRU WKH %HVW $OWHUQDWLYH�� (FRQRPHWULFD� ��

:HLW]PDQ� 0�� :� 1HZH\ DQG 0� 5DELQ �� ?6HTXHQWLDO 5' 6WUDWHJ\ IRU 6\QIXHOV�� %HOO

-RXUQDO RI (FRQRPLFV� ��

��

QDPLFV RI 1HZ 9HQWXUHV - Haas School of · PDF file9doxdwlrq dqg 5hwxuq ’\qdplfv ri 1hz...

Documents

Transcript of QDPLFV RI 1HZ 9HQWXUHV - Haas School of · PDF file9doxdwlrq dqg 5hwxuq ’\qdplfv ri 1hz...